Plunging motions of an elastically suspended wing with an oscillating flap an experimental and numerical assessment

Plunging motions of an elastically suspended wing with an oscillating flap An experimental and numerical assessment PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus Prof ir K.C.A.M Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op maandag 13 oktober 2014 om 10:00 uur door Joost Joachim Hermanus Marie STERENBORG ingenieur luchtvaart en ruimtevaart geboren te Nijmegen Dit proefschrift is goedgekeurd door de promotor: Prof dr ir drs H Bijl Copromotor: Dr ir A.H van Zuijlen Samenstelling promotiecommissie: Rector Magnificus Prof dr ir drs H Bijl Dr ir A.H van Zuijlen Prof dr ir G.A.M van Kuik Prof dr ir L.L.M Veldhuis Prof N.N Sørensen Dr -Ing Th Lutz Dr Ir K Boorsma Prof dr F Scarano voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft, copromotor Technische Universiteit Delft Technische Universiteit Delft Technical University of Denmark Universităat Stuttgart ECN Technische Universiteit Delft, reservelid This research is funded by Agentschap NL (formerly Senternovem), an agency of the Dutch Ministry of Economic Affairs, under project number EOS LT 09001 Printed by Ipskamp Drukkers, The Netherlands c 2014 by J.J.H.M Sterenborg Copyright All rights reserved No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without the prior permission of the author isbn: 978-94-6259-353-4 ii Summary Over the last years fluid-structure interactions have attained more interest emanating from applications, the availability of new numerical approaches for multi-physics coupling and the improved computing capacity that enables simulations of complex multi-physics problems Fluid-structure interactions involved in applications can be undesired but can also be benefited from An example of the latter is the popular research field of load alleviation for wind turbines based on aeroelastic blade deformations, like bend-twist coupling Next to aeroelastic load alleviation, active load mitigation systems for wind turbines also gain much attention Also for these active approaches, the aeroelastic system responses can be important to address Understanding and prediction of fluid-structure interactions can be achieved with numerical simulations One of the problems for numerical simulations of fluid-structure interactions is the validation of the solvers Main reason is the limited availability of proper experimental data, partly due to the complexity of experiments involving fluidstructure interactions This complexity appears amongst others in the determination of unsteady loads on moving objects and unsteady wind tunnel wall corrections Furthermore, the limited amount of data that are available are mostly for lower Reynolds regimes and/or different structures Also for aeroelastic codes used to design wind turbines there is a lack of validation data Therefore, this thesis aims to enhance the possibilities for validation of aeroelastic solvers used for the simulation of aeroelasticity of wind turbines An aeroelastic experiment is conducted using a wing based on the DU96-W-180 wind turbine profile and a Reynolds number of 700 000 Furthermore, in line with active load alleviation systems employing control surfaces, the one degree of freedom plunging wing motion is induced by controlled flap oscillations The flap actuation is sinusoidal as well as the resulting oscillations A rigid body motion is used in the experiment in order to eliminate spatial coupling between flow and structure in numerical simulations Three sub-objectives, elaborated on in the remainder, can be distinguished in this thesis: 1) the assessment of experimental unsteady load determination, 2) a one degree of freedom aeroelastic experiment to setup a validation database, and 3) a comparative numerical study using three 2-D aeroelastic solvers with different levels of fidelity, partially also to identify implications of the numerical modelling in combination with the experimental setup One option to determine the aerodynamic forces is to deduce the instantaneous sectional loads from measured velocity fields using Noca’s method along any closed contour An experiment is conducted to investigate the application of Noca’s method iii first for an aerofoil with an oscillating trailing edge flap Wind tunnel corrections for this specific unsteady flow problem are considered Conclusion of this assessment is that for the experimental data Noca’s approach is relatively sensitive to the contour location: applied to a set of contours a solution of the unsteady loads with an error bandwidth of on average 6.39% of its mean instantaneous values is found Also, compared to Kutta-Joukowski’s theorem and panel code simulations, small offsets of on average about 5% reduction are found in the force coefficients Among others, it is known a higher spatial resolution of the experimental data and more accurate approximations of velocity gradients will improve the force prediction Phase and amplitude of the lift are in close agreement with 2-D panel computations including modelled wind tunnel walls and a gap correction The aeroelastic experiment is conducted at an angle of attack of α = −0.95 ◦ yielding attached flow conditions The flap deflects over a range of about ±2 ◦ with reduced flap frequencies ranging from k = 0.1 to k = 0.3 The damped natural frequency of the mass-damper-spring like structure expressed as a reduced frequency is about k = 0.194 The obtained database contains displacements and time dependent aerodynamic forces It provides a clear insight in typical loads and motions and can be used in comparative studies As expected, the maximum displacement of the wing is found near the system eigenfrequency The lift is dominated by the flap motion and the effective angle of attack due to the motion introduces phase shifts of the lift signal with respect to the flap phase angle Despite the experiment has been setup and executed with the necessary precision, small ambiguities are found in the lift and drag and the data should not be used for code validation Structural assumptions (e.g mass-damper-spring, constant damping) are one of the causes for the ambiguities in the lift Both the data and experiences can be used to (re)design future experiments to improve the quality of the data to the desired level of accuracy for validation Suggestions in this are the extension of the used measurement techniques with surface pressure measurements and simplifications in the supporting structure by a reduction of the number of components In a comparative study the one degree of freedom aeroelastic problem is simulated with three different levels of fidelity 2-D aerodynamic models: Theodorsens model, a panel code and a URANS solver In the numerical models 2-D steady wind tunnel corrections are implemented All models are coupled to a structure solver and the fluid-structure interaction is resolved in both a loosely coupled and strongly coupled fashion The applicability of the 2-D wind tunnel corrections instead of a full modelling is investigated and the accuracy of the different models is assessed Trends in the lift forces, moments and displacements are predicted according to the experimental values, although some phase and amplitude errors are observed Errors are amongst others due to inherent 3-D flow effects in the experiment against 2-D simulations and the application of steady wind tunnel corrections on an unsteady problem Subiterations to reduce the coupling error only have a significant effect on the phase of the lift General conclusion is that compared to expensive 3-D simulations, less expensive 2-D solutions are found that approximate the experimental values for the unsteady test cases For Theodorsens model and the panel code this is achieved with a low computational effort and for URANS the computational effort is moderate iv Dompende bewegingen van een flexibel opgehangen vleugel met een oscillerende klep Een experimenteel en numeriek onderzoek Samenvatting De laatste jaren is vloeistof-vaste stof interactie onderwerp van onderzoek vanwege de vele toepassingen, de beschikbaarheid van nieuwe numerieke benaderingen voor multi-fysische koppelingen en de verbeterde rekenkracht die het mogelijk maakt om complexe, multi-fysische problemen door te rekenen Vloeistof-vaste stof interacties in toepassingen zijn soms onwenselijk, maar kunnen ook benut worden Een voorbeeld van het laatste kan worden gevonden in belastingsreducties voor windturbinebladen door middel van bladvervormingen onder invloed van luchtkrachten, zoals een buig-torsie koppeling Hiernaast is er veel belangstelling voor actieve systemen die belastingen reduceren op windturbinebladen Voor deze actieve systemen kan het ook belangrijk zijn om de vloeistof-vaste stof interactie te beschouwen Begrip van vloeistof-vaste stof interactie kan worden verkregen door numerieke simulaties uit te voeren Een van de problemen van numerieke simulaties is het valideren van de rekencodes Belangrijkste reden is de erg gelimiteerde beschikbaarheid van goede data, mede vanwege de complexiteit van experimenten Deze complexiteit komt onder andere voort uit de bepaling van de niet-stationaire krachten en instationaire windtunnelcorrecties Daarnaast is de gelimiteerde data vaak alleen beschikbaar voor lage Reynoldsgetallen en/of voor andere constructies Ook voor aero-elastische codes gebruikt voor het ontwerpen van windturbines is een gebrek aan validatie data Daarom is het doel van dit onderzoek om de mogelijkheden te vergroten om validaties uit te voeren van aero-elastische codes gebruikt voor het simuleren van windturbines Een aero-elastisch experiment is uitgevoerd, waarbij een vleugel gebaseerd op het DU96-W-180 windturbine profiel en een Reynoldsgetal van 700 000 zijn gebruikt Daarnaast is, net zoals bij actieve belastingsreductiesystemen met kleppen, de op-en-neergaande beweging van de gebruikte stijve vleugel genduceerd door een opgelegde klep beweging De klep oscilleert in een sinus beweging en daardoor beweegt ook de vleugel in een zelfde patroon Het gebruik van een stijve vleugel zorgt er voor dat in numerieke simulaties de koppeling tussen de rekenroosters voor de vloeistof en de vaste stof buiten beschouwing gelaten kan worden Het onderzoek kan worden onderverdeeld in delen, die in het vervolg worden beschouwd: 1) onderzoek naar de experimentele bepaling van instationaire krachten, 2) het ´e´en vrijheids graden aero-elastische experiment voor het vergaren van validatie materiaal, en 3) een vergelijkende studie van drie numerieke rekenmodellen met een verschillende complexiteit, deels ook om de implicaties van de modellering en de experimentele opstelling te beschouwen Een optie voor het bepalen van instationaire luchtkrachten is door de instantane doorsnede krachten te bepalen uit snelheidsvelden middels Noca’s methode toegepast v op een gesloten lijn Een experiment is uitgevoerd om de toepasbaarheid van Noca’s methode voor een vleugel met een bewegende klep te onderzoeken Windtunnelcorrecties voor dit specifieke instationaire probleem zijn ook onderzocht Conclusie is dat, gegeven de experimentele data, Noca’s methode relatief gevoelig is voor de gekozen ligging van de gesloten lijn: voor meerdere gesloten lijnen is een schatting van de instationaire krachten met een bandbreedte van 6.39% van de gemiddelde instantane kracht bepaald Verder zijn er, in vergelijking met Kutta-Joukowksi’s theorie en panelen code simulaties, kleine afwijkingen met een gemiddelde verlaging van zon 5% gevonden in de krachtencoefficiăenten Het is bekend dat onder andere een hogere resolutie voor de ruimte discretisatie en een hogere orde benadering voor de snelheidsafgeleiden leiden tot een verbetering van de voorspelling van de krachten De fase en de amplitude van de liftkracht komen goed overeen met 2-D simulaties met een panelen code met gentegreerde windtunnelcorrecties en een sleuf correctie Het aero-elastische experiment is uitgevoerd voor een invalshoek van α = −0.95 ◦, waarbij de stroming aanligt Hierbij slaat de klep uit over ±2 ◦ met gereduceerde flap frequenties van k = 0.1 tot k = 0.3 De gedempte natuurlijke frequentie van de massademper-veersysteem achtige constructie, uitgedrukt als een gereduceerde frequentie, is ongeveer k = 0.194 Verplaatsingen en tijdsafhankelijke luchtkrachten zijn gemeten De resultaten geven een goed inzicht in de typische krachten en verplaatsingen die kunnen worden gebruikt om vergelijkende onderzoeken te kunnen doen Zoals verwacht is de verplaatsing van de vleugel maximaal rond de natuurlijke frequentie van de constructie De liftkracht is met name afhankelijk van de klepbeweging De effectieve invalshoek door de verticale beweging zorgt voor fase veranderingen in de liftkracht ten opzichte van de klep fasehoek Ondanks een zorgvuldige opzet en uitvoering van het experiment, zijn er tegenstrijdigheden gevonden in de lift en weerstandskrachten die er toe leiden dat de data niet direct voor validatie kan worden gebruikt Aannames voor het structurele model (o.a massa-demper-veersysteem, constante demping) is een van de oorzaken voor de tegenstrijdigheden voor de liftkrachten De gemeten data en opgedane kennis kunnen worden gebruikt om een herontwerp te maken voor nieuwe experimenten, zodat nieuwe data geschikt voor validatie kan worden gemeten Suggesties hiervoor zijn het uitvoeren van oppervlakte drukmetingen en vereenvoudigingen van de ondersteunende constructie door minder componenten te gebruiken In een vergelijkingsonderzoek is het aero-elastische probleem gesimuleerd met drie 2-D stromingsmodellen van verschillende complexiteit: Theodorsens model, een panelen code en een URANS code In de simulatiemodellen zijn 2-D stationaire windtunnelcorrecties gentegreerd Alle stromingsmodellen zijn gekoppeld aan een structureel rekenmodel en de vloeistof-vaste stof interactie is zowel zwak en sterk gekoppeld opgelost De toepasbaarheid van 2-D windtunnelcorrecties in plaats van het simuleren van de volledige opstelling met windtunnel is onderzocht en de nauwkeurigheid van de rekenmodellen is beschouwd Trends in the liftkrachten, momenten en verplaatsingen zijn overeenkomstig voorspeld met de experimentele data, alhoewel fase- en amplitudefouten zijn waargenomen Fouten komen onder andere door de 2-D modellering van een 3-D experiment en de toepassing van stationaraire windtunnelcorrecties voor een instationair probleem Subiteraties om de koppelingsfouten te reduceren hebben alleen een waarneembaar effect op de fase van de liftkracht De algemene conclusie is dat ten opzicht van dure 3-D simulaties, minder dure 2-D voorspellingen zijn gevon- vi den die de resultaten van het experiment benaderen voor de instationaire problemen Voor Theodorsens model en de panelen code is dit bereikt met weinig rekentijd en voor URANS is de rekentijd meer gemiddeld vii viii Contents Summary iii Samenvatting v Contents xii Introduction 1.1 Motivation 1.2 Literature review of present state 1.3 Approach 1.4 Outline 1 Terminology, wind tunnel models and methodologies 2.1 Terminology 2.1.1 Characteristic (non-)dimensional numbers 2.1.2 Flap phase angle 2.1.3 Averaging methods 2.1.4 Data reduction 2.2 Wind tunnel model 2.2.1 Model description 2.2.2 Wing model derived with co-kriging 2.3 Standard wind tunnel corrections for steady flow 2.3.1 Steady corrections closed wind tunnels 2.3.2 Steady corrections for open jet wind tunnels 2.4 Wind tunnel measurements and FSI 2.5 Particle Image Velocimetry 2.5.1 Principles of PIV 2.5.2 Phase-locked PIV 2.6 Methods to derive (un)steady forces 2.6.1 Kutta-Joukowski’s circulatory approach 2.6.2 Noca’s momentum flux equation 2.6.3 Implementation of Noca’s method 9 10 10 11 11 11 12 13 13 14 15 16 16 17 17 17 18 20 ix 2.7 Uncertainty analysis Experimental benchmark I and numerical comparison: with actuated flap 3.1 Problem definition 3.1.1 Low turbulence tunnel 3.1.2 The model and equipment 3.1.3 Steady and unsteady test cases 3.1.4 PIV setup and apparatus 3.2 Numerical model 3.2.1 2-D panel code 3.3 Wind tunnel corrections 3.3.1 Wind tunnel wall corrections 3.3.2 Gap correction 3.4 Results 3.4.1 Force evaluation 3.4.2 Steady cases 3.4.3 Unsteady cases 3.5 Conclusions an aerofoil Experimental benchmark II: a free plunging wing with imposed oscillations 4.1 Problem definition 4.1.1 Open jet wind tunnel 4.1.2 Wind tunnel model 4.1.3 Supporting structure 4.1.4 Structural characteristics 4.1.5 Steady and unsteady test cases 4.2 Measurements and post-processing 4.2.1 Measurement devices 4.2.2 Force derivation 4.2.3 Measurement procedure and post-processing 4.2.4 Uncertainty analysis 4.2.5 PIV setup and apparatus 4.2.6 Wind tunnel corrections 4.3 Results 4.3.1 Steady cases 4.3.2 Unsteady case: 2-D PIV 4.3.3 Unsteady cases 4.4 Conclusions Comparative study of numerical models with imposed flap oscillations 5.1 Problem definition 5.1.1 Test problem simplification 5.1.2 Wind tunnel corrections 5.2 Numerical methods x 24 25 26 26 27 28 29 31 32 32 33 37 38 38 38 40 45 flap 47 48 48 49 49 50 52 54 54 55 57 58 59 61 61 61 67 70 80 for a free plunging aerofoil 83 83 83 84 88 Appendix H Wind tunnel measurements DU96-W-180 The wind tunnel model used in this work is based on the DU96-W-180 aerofoil It is an aerofoil that is part of the DU series, which consists of wind turbine dedicated aerofoils designed at Delft University of Technology The used wind tunnel model has manufacturing tolerances in the order of a few millimetres and therefore the model deviates significantly from the original DU96-W-180 aerofoil, as laid down in Appendix D Furthermore, on the wind tunnel model tripping wire is applied that has some impact on the flow and the determined force coefficients In this appendix 2-D steady wind tunnel measurements for the DU96-W-180 are presented along with integral, 2-D steady measurements for the current wind tunnel model in the same setup as discussed in Chapter Also graphs are provided where both data sets are plotted together H.1 Steady measurement data original DU96-W180 The DU aerofoils are tested in the wind tunnels of Delft University of Technology and IAG Stuttgart, as described by Timmer and van Rooij [2003] In this work the DU96-W-180 aerofoil is tested at a Reynolds number of Re=700 000, for which data is presented by Timmer [2010] Since no tabulated data is published previously for the DU96-W-180 at Re=700 000, in Table H.1 a selection of this data is presented, viz for −15 ◦ ≤ α ≤ 30 ◦ including some hysteresis targeted repeating measurements 147 α [ ◦] -15 -14.5 -14 -13.5 -13.04 -14.04 -15.03 -15.5 -13.04 -12.04 -11.04 -10.04 -9.04 -8.03 -7.03 -6.02 -5.02 -4.02 -3.01 -2.01 -1.00 0.00 1.01 2.01 3.02 4.02 5.03 6.03 cl -0.3348 -0.3458 -0.4003 -0.4964 -0.7263 -0.7198 -0.6964 -0.3402 -0.7263 -0.7152 -0.6830 -0.6486 -0.5933 -0.5201 -0.4340 -0.3429 -0.2478 -0.1467 -0.0387 0.0724 0.1824 0.2944 0.4053 0.5163 0.6272 0.7341 0.8430 0.9478 cd 0.1860 0.1808 0.1802 0.1716 0.0497 0.0767 0.1066 0.1937 0.0497 0.0361 0.0278 0.0228 0.0194 0.0165 0.0144 0.0129 0.0109 0.0097 0.0096 0.0097 0.0102 0.0109 0.0119 0.0130 0.0142 0.0159 0.0175 0.0196 α[ ◦] 7.03 8.04 8.54 9.04 9.54 10.05 11.04 12.04 13.04 14.04 15.04 16.04 17.04 18.04 19.04 20.03 21.03 22.03 23 22 21 20.03 23 24 26 28 30 cm 0.0204 0.0205 0.0222 0.0170 -0.0518 -0.0304 -0.0156 0.0222 -0.0518 -0.0605 -0.0645 -0.0630 -0.0606 -0.0585 -0.0576 -0.0571 -0.0565 -0.0564 -0.0569 -0.0577 -0.0589 -0.0600 -0.0612 -0.0623 -0.0634 -0.0639 -0.0648 -0.0652 cl 1.0546 1.1524 1.1959 1.2324 1.2441 1.2358 1.1847 1.1317 1.1102 1.0932 1.0872 1.0887 1.0864 1.0810 1.0668 1.0467 1.0188 1.0099 0.7755 0.7779 0.7902 1.0467 0.7755 0.7901 0.8838 0.9594 1.0193 cd 0.0218 0.0248 0.0254 0.0262 0.0283 0.0323 0.0503 0.0678 0.0776 0.0917 0.1038 0.1204 0.1353 0.1528 0.1695 0.1875 0.2061 0.2428 0.3612 0.3414 0.3255 0.1875 0.3612 0.4023 0.4823 0.5608 0.6377 cm -0.0660 -0.0658 -0.0640 -0.0613 -0.0559 -0.0511 -0.0553 -0.0572 -0.0524 -0.0535 -0.0534 -0.0567 -0.0598 -0.0637 -0.0674 -0.0726 -0.0773 -0.0894 -0.1179 -0.1171 -0.1167 -0.0726 -0.1179 -0.1456 -0.1666 -0.1864 -0.2050 Table H.1: Steady easurements DU-96-W180, Re=700 000 and −15 ◦ ≤ α ≤ 30 ◦ , clean configuration Source: N Timmer H.2 Steady measurement data current model Steady force measurements for the current wind tunnel model with tripping wires are briefly discussed in this section Tripping wires are applied for all wind tunnel tests presented in this thesis The tripping wires are used to have more confidence on the location of transition, especially also in conjunction with the oscillating flap and the changing pressures In Table H.2 the steady quasi 2-D measurements are presented for tripped conditions The measurements are quasi 2-D, since balance measurements are conducted for the model including the gaps of about mm between the model and the wind tunnel walls These gaps introduce three dimensional flow The measured data is corrected with the wind tunnel measurement system incorporated correction procedure presented in Appendix A 148 α [ ◦] -5.14 -4.11 -3.07 -2.04 -1.01 0.03 1.06 2.1 3.13 4.17 5.2 6.23 7.25 8.27 9.28 10.28 11.29 12.3 13.31 14.31 15.31 16.3 17.3 18.29 19.28 20.26 21.25 22.23 23.11 24.11 25.1 cl -0.168 -0.083 0.007 0.09 0.175 0.263 0.352 0.44 0.526 0.607 0.684 0.751 0.806 0.84 0.865 0.896 0.92 0.95 0.967 0.985 0.997 0.991 0.978 0.98 0.974 0.952 0.933 0.921 0.724 0.737 0.74 cd 0.01447 0.01394 0.01377 0.01459 0.01575 0.01724 0.01867 0.02036 0.02208 0.02425 0.02658 0.02954 0.03321 0.03895 0.04797 0.05724 0.0656 0.07435 0.08412 0.09349 0.10363 0.1189 0.13181 0.14734 0.16551 0.18423 0.20407 0.22672 0.34415 0.36347 0.38094 cm -0.05773 -0.05551 -0.05346 -0.0502 -0.04755 -0.04502 -0.04317 -0.04052 -0.03827 -0.03506 -0.03164 -0.02747 -0.0232 -0.02103 -0.02168 -0.02389 -0.02358 -0.02601 -0.02607 -0.02823 -0.0292 -0.03414 -0.03667 -0.0418 -0.04629 -0.05273 -0.05911 -0.06791 -0.10338 -0.10701 -0.11196 Table H.2: Balance measurements current model, Re=700 000 and −5.14 ◦ ≤ α ≤ 25.1 ◦ , tripped configuration 149 H.3 Combined graphs DU96-W-180 and current wind tunnel model To provide more insight, the data of Table H.2 are plotted together with part of the data of Table H.1 in Figure H.1 Freestream lift, drag and moments coefficients for both the clean DU96-W-180 aerofoil and the tripped wind tunnel model are shown, without the hysteresis loop It can clearly be seen that the data of both wind tunnel models differs significantly As explained the different geometry, the use of tripping wires and three dimensional flow due to the gaps are main causes for the observed differences It is not possible to determine the single contributions for each of these causes from the presented data More wind tunnel tests are needed to accomplish this The trends in the lift and the drag are according to expectancy: 1) a decrease of the lift due to three dimensional flow and also possible decambering due to the tripping wire and shape deviation (see Appendix D), and 2) an increase in drag due to the same contributions For the moment coefficient the trend is more difficult to interpret and pressure measurements would be very helpful 1.4 1.2 cl , cL [−] 0.8 0.6 0.4 0.2 -0.2 DU96-W-180 (clean) Used model (tripped) -0.4 -8 -4 12 16 α[ ◦ ] (a) 2-D and integral lift coefficients 150 20 24 28 0.45 DU96-W-180 (clean) Used model (tripped) 0.4 0.35 cd , cD [−] 0.3 0.25 0.2 0.15 0.1 0.05 -8 -4 12 16 20 24 28 20 24 28 α[ ◦ ] (b) 2-D and integral drag coefficients -0.02 -0.04 cm , cM [−] -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 DU96-W-180 (clean) Used model (tripped) -0.18 -8 -4 12 16 α[ ◦ ] (c) 2-D and integral moment coefficient around 0.25c Figure H.1: Comparison of lift, drag and moment coefficients for DU96-W-180 (2-D, clean) and used wind tunnel model based on DU96-W-180 (integral, tripped, with slits near tunnel walls) Excluding hysteresis loop Source DU96-W-180 (2-D, clean): N Timmer 151 152 Appendix I Tabulated results grid and time step study (U)RANS solver In Chapter a comparative study is reported using various numerical models Standard practise in numerical computations is to conduct a verification study, see e.g Roache [1994] and Stern et al [2001] Chapter provides an overview of the final results; in this appendix the intermediate solutions are given of the grid and time step study for the (U)RANS solver The post-processing is applied as described in Section 2.1.4 For a fair convergence study the maximum level of accuracy is rounded up to four digits, to take into account also inaccuracies that are introduced due to post-processing steps in the data The grid convergence study for the RANS solver is based on steady calculations employing three, body conformal unstructured meshes The meshes have 75k cells, 130k cells and 211k cells In Table I.1 the lift, drag and moment coefficients are presented for each of the meshes Table I.1: Spatial convergence study RANS using three meshes: 75k, 130k and 211k grid cells cl cd cm 75k mesh 0.2924 0.0197 −0.0587 130k mesh 0.2774 0.0172 −0.0553 211k mesh 0.2716 0.0160 −0.0541 The time step study for the URANS solver is conducted using the 130k mesh, applied to the test case with k = 0.198 and δamp = 1.70 ◦ For three time steps, viz ∆t = 0.01 s, ∆t = 0.005 s and ∆t = 0.0025 s, the results for the lift, drag and moment coefficient are provided in Table I.2 Mind that the mean results differ compared to the steady simulations, since the unsteady simulations are conducted including wind tunnel corrections This should however not influence the convergence 153 Table I.2: Temporal convergence study URANS using the 130k mesh with three time steps: ∆t = 0.01 s, ∆t = 0.005 s and ∆t = 0.0025 s Including open 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[2010] and Appendix H The wing is untwisted and untapered and has a chord of 0.5 m, a span of 1.8 m and a 0.2c flap Except for the wing span, which is based on wind tunnel dimensions, the wing. .. 0.1 the flap mean deflection δmean and the amplitude δamp are set to ◦ and ◦ respectively For k = 0.2, next to these settings also a mean deflection of ◦ and an amplitude of ◦ are used Standard... corrections, the following steady and unsteady cases are presented: k = with δmean = ◦ or ◦ ; k = 0.1 with δmean = ◦ and δamp = ◦ ; k = 0.2, δmean = ◦ and δamp = ◦ ; k = 0.2, δmean = ◦ and δamp = ◦