Diploma work. University diploma study programme of Mechanical Engineering Environmental Engineering. Prof. Dr. Leopold Skerget

Transcription

1 AERODYNAMIC ANALYSIS OF AN OSCILLATING AIRFOIL IN A POWER-EXTRACTION REGIME USING THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH EFFICIENT AND ACCURATE LOW-SPEED PRECONDITIONING Student: Study programme: Module: Jernej DROFELNIK University diploma study programme of Mechanical Engineering Environmental Engineering Supervisor: Co-supervisor: Prof. Dr. Leopold Skerget Dr. M. Sergio Campobasso Maribor, July 2012

2 AERODINAMIČNA ANALIZA NIHAJOČEGA KRILA V ENERGIJSKO-EKSTRAKCIJSKEM REŽIMU Z UPORABO STISLJIVIH NAVIER-STOKESOVIH ENAČB Z UČINKOVITIM IN NATANČNIM NIZKO-HITROSTNIM PREDPOGOJEVALNIKOM Diplomsko delo Študent: Študijski program: Smer: Jernej DROFELNIK Univerzitetni študijski program Strojništvo Okoljevarstveno inženirstvo Mentor: Somentor: red. prof. dr. Leopold Škerget Dr. M. Sergio Campobasso Maribor, julij 2012

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4 IZJAVA Podpisani Jernej Drofelnik izjavljam, da: je predložena disertacija samostojno delo, ki je bilo opravljeno pod mentorstvom red. prof. dr. Leopolda Škergeta in somentorstvom Dr. M. Sergia Campobassa; predložena disertacija v celoti ali v delih ni bila predložena za pridobitev kakršnekoli izobrazbe na drugi fakulteti ali univerzi; soglašam z javno dostopnostjo disertacije v Knjižnici tehniških fakultet Univerze v Mariboru. Datum:, Podpis: II

5 ACKNOWLEDGMENTS It is a pleasure to thank the many people who made this diploma work possible. I would like to thank my supervisor Prof. Dr. Leopold Skerget from the Faculty of Mechanical Engineering at Maribor University, for his support and guidance. My sincere gratitude goes to my co-supervisor Dr. M. Sergio Campobasso from the School of Engineering at Glasgow University, for accepting me to the summer internship in his research group, for the continuous support of my research, for his patience, motivation, enthusiasm, and immense knowledge. I wish to thank my entire extended family and closest friends for providing a loving environment for me. I would like to thank my dear girlfriend Teja, who was always there cheering me up and stood by me through the good times and bad. Special thank goes to my parents, they were always supporting me and encouraging me with their best wishes. III

6 AERODYNAMIC ANALYSIS OF AN OSCILLATING AIRFOIL IN A POWER-EXTRACTION REGIME USING THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH EFFICIENT AND ACCURATE LOW-SPEED PRECONDITIONING UDK: (043.2) Keywords: Low-speed preconditioning, compressible multigrid Navier-Stokes solver, energy-extracting oscillating wing, computational fluid dynamics. Abstract A wing that is simultaneously heaving and pitching may extract energy from an oncoming air flow, thus acting as turbine. The purpose of this study was to analyse the relationship between the aerodynamics and the theoretical performance of this device by means of timedependent laminar flow simulations performed with a research compressible finite volume Navier-Stokes solver COSA. In presented analyses, which confirm the findings of another independent study, the efficiency of the power extraction of this device can be of the order of 35 %, and such an efficient operating condition is characterized due to the favourable effects of a strong dynamic stall. This study is a part of a wider research programme of Dr. Campobasso s group at University of Glasgow, aiming to develop a general-purpose computational framework for unsteady aerodynamic and aeroacoustic wind energy engineering. In view of aeroacoustic applications, the developed flow solver uses the compressible formulation of the Navier- Stokes equations with carefully optimized low-speed preconditioning. To demonstrate the modeling capabilities, the accuracy and the high computational performance of the developed low-speed preconditioning technology, the unsteady aerodynamics of the energyextracting device is simulated by using a computationally challenging freestream Mach number of A mixed preconditioning strategy that maintains both the nominal accuracy and the computational efficiency of the solver also for time-dependent low-speed problems is presented. A fundamental element of novelty of this study is a thorough assessment of the proposed approach partly based on the challenging and realistic problem associated with the oscillating wing device. IV

7 AERODINAMIČNA ANALIZA NIHAJOČEGA KRILA V ENERGIJSKO-EKSTRAKCIJSKEM REŽIMU Z UPORABO STISLJIVIH NAVIER-STOKESOVIH ENAČB Z UČINKOVITIM IN NATANČNIM NIZKO-HITROSTNIM PREDPOGOJEVALNIKOM UDK: (043.2) Ključne besede: Nizko-hitrostni predpogojevalnik, večmrežna metoda, Navier-Stokesov programski paket za stisljivo tekočino, energijsko-ekstrakcijsko nihajoče krilo, računalniška dinamika tekočin. Povzetek V zračnem toku nihajoče krilo, ki se sočasno nagiba okrog prečne osi, lahko deluje kot turbina. Namen študije je analiza odvisnosti aerodinamike in teoretične zmogljivosti te naprave s pomočjo časovno-odvisnih laminarnih tokovnih simulacij, izvedenih z akademskim programskim paketom COSA, ki uporablja formulacijo Navier-Stokesovih enačb nestacionarnega gibanja newtonske viskozne stisljive tekočine na osnovi metode končnih volumnov. Analiza potrjuje ugotovitve druge neodvisne študije, da lahko ta naprava dosega izkoristke v območju 35 %. Dober izkoristek je posledica ugodnih učinkov močnega odtrganja zračnega toka iz krila. Študija je del širšega raziskovalnega programa, ki ga vodi Dr. Campobasso na Univerzi v Glasgowu, ki razvija splošen programski sestav za reševanje problemov nestacionarne aerodinamike in aeroakustike vetrno-energijskega inženiringa. Glede na aeroakustične aplikacije uporablja COSA Navier-Stokesove enačbe za stisljivo tekočino s skrbno oblikovanim nizko-hitrostnim predpogojevalnikom. Za demonstracijo zmožnosti, natančnosti in visoke računske zmogljivosti razvite nizko-hitrostne predpogojevalne tehnologije, je bila simulirana nestacionarna aerodinamika energijsko-ekstrakcijske naprave, z uporabo zahtevnega prosto-tokovnega Machovega števila Predstavljena je strategija mešanega nizkohitrostnega predpogojevalnika, ki vzdržuje nominalno natančnost in računsko učinkovitost programskega paketa tudi za časovno-odvisne nizko-hitrostne probleme. Bistvena novost je temeljita ocena predlaganega pristopa, delno zasnovanega na zahtevnem in realističnem problemu, povezanem z napravo z nihajočim krilom. V

8 V diplomski nalogi je prikazana aerodinamična analiza nihajočega krila, ki odvzema energijo vetru in jo pretvarja v mehansko delo. Predvsem smo se osredotočili na procese nestacionarne aerodinamike, ki pripomorejo k večjemu izkoristku te naprave. Dodatno so predstavljeni tudi v programski paket COSA implementirani algoritmi za nizko-hitrostni predpogojevalnik in demonstracija njihovega delovanja. Za demonstracijo zmožnosti in učinkovitosti programskega paketa z nizko-hitrostnim predpogojevalnikom, so bili uporabljeni standardni testni primeri, na koncu pa še aerodinamično bolj kompleksen primer nihajočega krila. V prvem poglavju so najprej opisani prenosni pojavi v tekočinah z osnovnimi fizikalnimi zakoni o ohranitvi mase, gibalne količine in energije v integralski obliki za kontrolni volumen. Nato so ti zakoni zapisani še v sistemu Navier-Stokesovih enačb in v nadaljevanju tudi v diferencialni obliki. Prav tako je predstavljeno, kakšen sistem nelinearnih parcialnih diferencialnih enačb rešuje COSA. Ker COSA uporablja krivočrtni koordinatni sistem, je opisana pretvorba koordinat kartezičnega koordinatnega sistema v krivočrtni koordinatni sistem z uporabo preslikave, ki je lokalno inverzibilna v vsaki točki. V naslednjem poglavju je nato opisana krajevna diskretizacija in integracija časovno-odvisnih enačb. Temu sledi še zgradba nizko-hitrostnega predpogojevalnika za stacionarne in nestacionarne razmere tokovnega polja. Šesto poglavje je namenjeno opisu in delovanju naprave z nihajočim krilom. Z gibalnimi enačbami je opisano vsiljeno gibanje te naprave. Prav tako je prikazano kako na pridobivanje energije vpliva kot zamika faze φ in kakšen je vpliv efektivnega vpadnega kota α, na obratovalni režim naprave. Naprava pridobiva energijo, ko je α(t/4) < 0, v nasprotnem primeru, ko je α(t/4) > 0, pa deluje kot pogonsko sredstvo. Sledi poglavje o validaciji natančnosti programskega paketa COSA. Najprej je primerjana stacionarna simulacija toka vzdolž ravne plošče z Blasiusovo analitično rešitvijo. Sledi analiza zmogljivosti nizko-hitrostnega predpogojevalnika za tri različna Machova števila (M = 0.1, M = 0.01 in M = 0.001) in primerjava računske učinkovitosti programskega paketa za primer uporabe in neuporabe nizko-hitrostnega predpogojevalnika. Nato sledi še testni primer obtekanja valja z von Karmanovo vrtinčno stezo, ki potrjuje natančnost programskega paketa drugega reda. V naslednjem poglavju, ki je razdeljeno na dva dela, so podani rezultati. Najprej je predstavljena aerodinamična analiza nihajočega krila za dva različna obratovalna režima in sicer za režim visoke učinkovitosti odvzemanja energije zračnemu toku (primer A) in režim nizke učinkovitosti odvzemanja energije zračnemu toku (primer B). Razložen je proces vrtinčenja sprednjega roba aerodinamičnega profila (LEVS) in opis VI

9 potovanja vrtinca od sprednjega do zadnjega roba aerodinamičnega profila do trenutka, ko vrtinec profil zapusti. Ta pojav zelo ugodno vpliva na obratovalni režim visoke učinkovitosti odvzemanja energije (primer A). Drugi del poglavja je namenjen analizi zmogljivosti in natančnosti nizko-hitrostnega predpogojevalnika za primer nihajočega krila. Opisanih je šest simulacij za tri različna Machova števila (M = 0.1, M = 0.01 in M = 0.001), za primer uporabe in neuporabe nizko-hitrostnega predpogojevalnika. Za vsako Machovo število, v primeru uporabe in neuporabe nizko-hitrostnega predpogojevalnika, je primerjana natančnost rezultatov koeficienta tlaka in vrtinčnega polja ter računska učinkovitost predpogojevalne tehnologije. Zadnje poglavje je kratek pregled najbolj pomembnih rezultatov in ugotovitev. VII

10 CONTENTS 1 INTRODUCTION 1 2 GOVERNING EQUATIONS INTEGRAL FORM CONSERVATION OF MASS CONSERVATION OF MOMENTUM CONSERVATION OF ENERGY NAVIER-STOKES EQUATIONS FORMULATION FOR A PERFECT GAS DIFFERENTIAL FORM COSA GENERALIZED CURVILINEAR COORDINATES 13 4 CFD SOLVER SPACE DISCRETIZATION NUMERICAL INTEGRATION OF TIME-DEPENDENT EQUATIONS LOW-SPEED PRECONDITIONING 19 6 OSCILLATING WING DEVICE 22 7 VALIDATION LAMINAR FLAT PLATE VORTEX SHEDDING OF CIRCULAR CYLINDER RESULTS AERODYNAMIC ANALYSIS PERFORMANCE OF LOW-SPEED PRECONDITIONING CONCLUSIONS 47 A GENERALIZED CURVILINEAR COORDINATES 52 VIII

11 NOMENCLATURE ACRONYMS 2D 3D AoA CFD CFL Two-dimensional Three-dimensional Angle of attack Computational fluid dynamics Courant-Friedrichs-Lewy number FERK Fully-explicit Runge-Kutta HAWT Horizontal-axis wind turbines IRS LE Implicit residual smoothing Leading edge LEVS Leading edge vortex shedding LTS MG MP NP NS Local time-stepping Multigrid Mixed preconditioning No preconditioning Navier-Stokes PDE s Partial differential equations PIRK Point-implicit Runge-Kutta RANS Reynolds-averaged Navier-Stokes equations RK Runge-Kutta RMS Root mean square TE Trailing edge IX

12 VAWT Vertical-axis wind turbines VP Viscous preconditioning SYMBOLS Transpose operator α t q h η η w γ κ F i F exact Effective angle of attack Physical time step Time rate of heat transfer per unit mass Efficiency Nondimensional wall distance Ratio of specific heat coefficients Thermal diffusivity coefficient Integral output variable of the CFD simulation Unknown exact value of the considered output µ Dynamic viscosity µ re f Reference dynamic viscosity Ω ω Ω f Ω f σ τ s Kinematic viscosity Pitching velocity Angular frequency Vorticity Vorticity normalized by the freestream velocity Normal stress tensor Viscous stress tensor Mean strain rate tensor X

13 φ ρ τ θ 0 Φ i Φ v F C F D n Q S R v C c C f C M C P C p c p c s c v C X C Y Phase angle Fluid density Pseudo time Pitching amplitude Inviscid flux vector Viscous flux vector Convective flux vector Diffusive flux vector Unit normal vector Surface sources Resultant force Flow velocity vector Control volume Airfoil chord Skin-friction coefficient Torque coefficient Power coefficient Pressure coefficient Specific heat coefficient under constant pressure Local speed of sound Specific heat coefficient under constant volume Horizontal force coefficient Vertical (heaving) force coefficient XI

14 d d c E e f f f e f MG f sh H h h 0 I k l r L IRS M N T N pde P p P θ P a Overall vertical extent of the airfoil motion Cylinder diameter Total energy Internal energy per unit mass Vibration frequency Nondimensionalized frequency Body force per unit mass Multigrid forcing function Frequency of vortex shedding Total enthalpy Enthalpy per unit mass Heaving amplitude Kronecker delta function Thermal conductivity coefficient The logarithm with base 10 of the RMS of all cell-residuals Implicit residual smoothing operator Torque Approximate number of time-intervals per period Nonlinear partial differential equations Total extracted power Pressure Pitching power contribution Total available power of the oncoming flow XII

15 P y Q Q s r R g R Φ S St T T S T re f U u V V c v e v y x p Heaving power contribution Unknown flow variables Source term Gas constant per unit mass Residual vector Cell residuals Control surface Strouhal number Absolute temperature Sutherland temperature Reference temperature Conservative flow variables Freestream velocity Volumes of the grid cells Contravariant velocity Effective velocity Heaving velocity Pitching center XIII

16 1 INTRODUCTION Mankind is always searching for new, efficient and environmentally friendly energy-conversion devices. Recently the use of renewable energy sources and zero-emission technology has become widely popular. In aeronautics, the phenomenon of self-feeding and potentially destructive vibration, known as wing flutter is a well established concept. Under suitable conditions an aircraft wing may absorb energy from the air flow, however, consequences for the aircraft can be catastrophic, the wing will begin to oscillate, pitching and plunging. If this occurs an aircraft during it s flight, the structure of the wing material can be damaged. However, an airfoil which is mechanically coupled in pitch and plunge, it can extract the energy from an oncoming airflow without harming itself. Nowadays, a great deal of electric energy production from the wind energy is mainly based on the use of multi-megawatt horizontal-axis wind turbines. Small horizontal- and verticalaxis wind turbines are currently being developed for small-scale energy production, the power achieved of these devices is estimated of the order 10kW, and are typically installed both in rural and urban areas. Small horizontal- and, to a minor extent, vertical-axis wind turbines have so far dominated the market of small-scale wind machines. However, in this diploma work, the use of significantly different devices for power generation from the wind energy is presented. Oscillating wing device could be used in the field of wind, hydro and tidal energy systems, relying on the use of oscillating wings simultaneously heaving and pitching. In 1981, McKinney and Delaurier [19], built an experimental device and named it wingmill. The proposed concept was tested in a wind tunnel and they claimed it could achieve performance levels competitive with conventional windmills. In 2003 Jones et al. [14] further investigated and performed a numerical analysis for flapping-wing propulsion and power generation using an unsteady, potential flow code. One of the most recent investigations of power-extraction device has been done by Kinsey and Dumas [16] in A thorough parametric computational fluid dynamics (CFD) investigation into the effects of motion, geometric and viscous parameters has been performed. It was reported that this device can extract energy from the wind with efficiencies as high as 35 %. The main aerodynamic feature responsible for such a high efficiency is the unsteady leading edge vortex shedding associated with dynamic stall. Their parametric study has also shown that motion parameters such as heaving amplitude, pitching amplitude and frequency primarily govern the predicted performances. The geometric parameters, such as airfoil shape and loca- 1

17 tion of pitching axis, and viscous parameters, which includes Reynolds number and turbulence modeling, do not have as much impact on performance. Using higher Reynolds number the performance in power-extraction mode slightly increases. This diploma work also considers the aerodynamic analysis of this device to further highlight the complex unsteady aerodynamics controlling the energy extraction process, and thus stimulate further numerical and experimental research into this promising device. The time-dependent two-dimensional (2D), unsteady laminar flow analyses reported in [16] have been performed using the incompressible Navier- Stokes (NS) flow solver of the commercial CFD package FLUENT. All computations have been carried out in the heaving reference frame of the airfoil, thus leaving only the pitching motion of the airfoil to be dealt with through a rigid-body mesh rotation and a circular nonconformal sliding interface. In this diploma work the aerodynamic analysis of the same device is performed adopting a substantially different approach such as those reported in [16]. The time-dependent 2D laminar flow simulations are carried out using a compressible NS solver, the research CFD code COSA [6, 5, 7] developed by Dr. Campobasso s group at the School of Engineering of Glasgow University. One further important difference with respect to the analyses of [16] is that in the present computational studies are performed in the absolute reference frame, thus making use of computational meshes simultaneously heaving and pitching. The level of Mach number at which wind energy devices are expected to operate is well below 0.3, the aerodynamic analysis can be performed by incompressible solvers, such as those in [16, 23]. An alternative approach is to use compressible solvers [18] with low-speed preconditioning (LSP) [25, 26]. When using a compressible solver for low-speed flows, the eigenvalues of the inviscid flux Jacobian differ from each other by several orders of magnitude. This difference between acoustic and convective speeds results in an imbalance of the amount of numerical dissipation in the flow equations, and this occurrence spoils the accuracy of the solution. The use of low-speed preconditioning results in a re-equalization of acoustic and convective speeds, which allows the nominal accuracy of the solver to be restored. When using iterative integration methods with a CFL constraint, the eigenvalue equalization also yields higher convergence rates, which at least based on the theoretical analyses made to design low-speed preconditioners and the numerical solution of relatively simple test cases, are independent of the freestream Mach number M. However, there are not many published studies on the use of the compressible NS equations with LSP to solve complex low-speed atmospheric problems which assess and discuss both the solution accuracy and the computational efficiency of the 2

18 preconditioned solver. When using an explicit integration procedure, such as the multigrid iteration implemented by COSA, for developing a LSP strategy that only maintains the solution accuracy for a wide range of realistic low-speed problems is simpler than developing a LSP strategy that also optimizes the convergence properties of the solver for the same wide class of flows. Maintaining a good computational performance of preconditioned explicit solvers is particularly difficult for time-dependent analyses. The computational efficiency of LSP for complex flow problems needs to be addressed, particularly in the view of attractiveness of the compressible formulation due to its potential of performing accurate aeroacoustic analyses. Another reason for analyzing the aerodynamics and the performance of the oscillating wing device is thus to demonstrate the predictive accuracy and the computational efficiency of the preconditioned compressible approach of the COSA solver by considering a complex and realistic wind energy problem. The main objective of this diploma work is detailed aerodynamic analysis of the oscillating wing device in power extraction from the wind, focusing on the unsteady aerodynamics features leading to a high efficiency of power conversion. On the other hand, it also summarizes the main algorithmic features of the low-speed preconditioner implemented by COSA, and demonstrates the predictive capabilities and high computational efficiency of the preconditioned solver by solving a set of relatively standard test cases and the aerodynamically more complex oscillating wing device animated by simultaneous heaving and pitching motion. The diploma work first presents, basic equations of fluid dynamics, following the governing equations for problems with moving grids, the main algorithmic features of the structured multi-block COSA solver and the LSP strategy implemented by this code. It then provides a description of the oscillating wing device and its main kinematic parameters, followed by a section on the validation of the preconditioned solver and the computational characteristics of the preconditioner. A detailed analysis of the power-extracting device is then reported, which highlights the effects of dynamic stall on the power extraction process, but also assesses the performance of the developed LSP methodology for this complex problem. A summary of the presented material and further prospective investigations are provided in the closing section. 3

19 2 GOVERNING EQUATIONS 2.1 INTEGRAL FORM The derivation of the key laws describing the evolution of fluid flows is based on the fact that the dynamical behaviour of a fluid is defined by the conservation of the following three quantities [3]: the conservation of mass, the conservation of momentum, the conservation of energy. The conservation of a certain flow quantity means that the variation of conserved flow quantity within a given volume is due to the net effect of some internal sources and of the amount of that quantity being transported across the boundary and any internal and external forces acting on the volume. This amount is called flux and can be in general decomposed into two different parts: first part is due to the convective transport and second part due to the molecular motion present in the fluid at rest. Second contribution is of a diffusive nature and it is proportional to the gradient of the quantity considered and hence it will vanish for a homogeneous distribution CONSERVATION OF MASS A fixed, non-deforming volume of fluid called the control volume C, which has defined surface boundary called control surface S is considered. If the attention to single-phase fluids is restricted, the law of mass conservation expresses the fact that mass cannot be created in such a fluid system, nor can disappear from it. There is also non-diffusive flux contribution to the continuity equation, since for a fluid at rest any variation of mass would imply a displacement of fluid particles. The general integral mass conservation equation, also called continuity equation is the following: t C ρdc+ ρ( v n)ds = 0, (1) S [ where ρ denote the fluid density, v is the flow velocity and is defined as v = u v w [ n is the unit normal vector defined as n = n x n y n z ], where the superscript denotes the transpose operator. 4 ],

20 2.1.2 CONSERVATION OF MOMENTUM Newton s second law states that the rate of change of momentum of a fluid particle equals the sum of the forces on the particle. Momentum is a vector quantity defined as the product of density and velocity ρ v, when expressed per unit of volume. The contribution of the convective flux tensor to the conservation of momentum is then given by: ρ v( v n)ds. S The diffusive flux is zero, since there is no diffusion of momentum possible for a fluid at rest. The contribution of the body force to the momentum conservation is: ρ f e dc, C where ρ f e is the body force per unit volume. The surface sources, the last segment of momentum conservation equation consists of two parts. First part is an isotropic pressure component, denoted by p and second part is a viscous stress tensor τ. The stress tensor show how normal and shear stresses are connected to the flow velocity. Viscous stress tensor is originated by the friction between the fluid and the surface of an element. The general form in Cartesian coordinates is given by: τ xx τ xy τ xz τ= τ yx τ yy τ yz. τ zx τ zy τ zz For the vast majority of practical problems, where the fluid can be assumed to be Newtonian, the components of the viscous stress tensor are defined by the relation: (2) τ=2µ[s 1/3( v)i], s=( v+ v )/2, (3) (4) where µ is the dynamic viscosity, s is the mean strain rate tensor, and I is the Kronecker delta function. Sutherland s Law is used to compute the dynamic viscosity µ, as function of the absolute temperature T of an ideal gas. It is based on kinetic theory of ideal gases and an idealized intermolecular-force potential. It can be expressed as: µ= µ re f ( T T re f ) 3/2 T re f + T S T + T S, (5) 5

21 where T re f and µ re f are the reference temperature and the reference dynamic viscosity, respectively and T S is the Sutherland temperature. Sutherland s Law coefficients are: µ re f = kgm 1 s 1,T re f = K,T S = K. To sum up all the above contributions according to the general conservation law, the expression for the momentum conservation is obtained inside an arbitrary control volume C which is fixed in space with control surface S: ρ vdc + ρ v( v n)ds = ρ f e dc p nds + (τ n)ds. (6) t C S C S S CONSERVATION OF ENERGY The conserved quantity in a fluid is the total energy defined as the sum of its internal energy per unit mass e and its kinetic energy per unit mass( v) 2 /2. The underlying principle that we will apply in the derivation of the energy equation, is the first law of thermodynamics. Applied to the control volume C, it states that any changes in time of the total energy inside the volume are caused by the rate of work of forces acting on the volume and by the net heat flux into it. The expression for the total energy is: E = e+ v2 2 = e+ u2 + v 2 + w 2. (7) 2 The conservative quantity is the total energy per unit volume ρe. The contribution of the convective flux F C is: F C = ρ v(e+ v2 2 ). The term diffusive flux F D is present here, in contrast to the continuity and the momentum equation. Since it is defined for fluid at rest, only the internal energy becomes effective and it is obtained: F D = γρκ e, (8) where the coefficient γ=c p /c v is the ratio of specific heat coefficients under constant pressure and constant volume, κ is the thermal diffusivity coefficient and has to be defined empirically. The diffusive term describes the diffusion of heat in a medium at rest due to molecular thermal conduction and is in general written in the form of Fourier s law of heat conduction: F D = k T, (9) 6

22 where k is the thermal conductivity coefficient and T is the absolute static temperature. The volume sources are the sum of the work of the volume forces f e and of the volumetric heating due to absorption or emission of radiation, or due to chemical reactions. Hence we have the volume sources: Q V = ρ f e v+ q h, (10) where q h is the time rate of heat transfer per unit mass. The surface sources Q S are the result of work done on the fluid by the pressure as well as the shear and the normal stresses acting on the surface of the volume considering that there are no external surface heat sources: Q S = σ v= p v+τ v, (11) where σ denotes the normal stress. Grouping all the contributions, the energy conservation equation can be written in the form: ρedc+ ρe( v n)ds = k( t T n)ds+ (ρ f e v+ q h )dc+ (σ v) nds. (12) C S S Clarifying the term (σ v), and introducing the enthalpy per unit mass of the fluid h = (e+ p ρ ), leads to the following alternative expression: ρedc+ ρh( v n)ds = k( t T n)ds+ (ρ f e v+ q h )dc+ (τ v) nds, (13) C S S C S where H is the total enthalpy, given by: C S H = h+ v2 2 = E+ p ρ, (14) NAVIER-STOKES EQUATIONS Conservation law of mass, momentum and energy can all be collected into one system of equations in order to obtain a better overview of the various terms involved. Two flux vectors are introduced, namely inviscid flux vector Φ i which is related to the convective transport of quantities in the fluid and vector of viscous fluxes Φ v, which contains the viscous stresses as well as the heat diffusion. Additionally a source term Q s is defined, it comprises all volume sources due to body forces and volumetric heating. Given a fixed control volume C with boundary S, the integral form of the 3D time-dependent Reynolds-averaged Navier-Stokes equations (RANS) is presented: UdC+ ( Φ i Φ v )ds= Q s dc. (15) t C S C 7

23 The array U of conservative flow variables, consists in three dimensions of the following five components: [ U = ρ ρu ρv ρw ρe ]. (16) The inviscid flux vector is given by the array Φ i : ρv c ρuv c + n x p Φ i = ρvv c + n y p, (17) ρwv c + n z p ρhv c with the contravariant velocity V c, which is the velocity normal to the surface element ds. It is defined as the scalar product of the velocity vector and the unit normal vector: V c = n x u+n y v+n z w. (18) The viscous flux vector is given by the array Φ v : 0 n x τ xx + n y τ xy + n z τ xz Φ v = n x τ yx + n y τ yy + n z τ yz, (19) n x τ zx + n y τ zy + n z τ zz n x Θ x + n y Θ y + n z Θ z where: Θ x = uτ xx + vτ xy + wτ xz + k T x, (20) Θ y = uτ yx + vτ yy + wτ yz + k T y, (21) Θ z = uτ zx + vτ zy + wτ zz + k T z, (22) are the terms describing the work of viscous stresses and the heat conduction in the fluid. 8

24 Finally the source term reads: 0 ρ f e,x Q s = ρ f e,y. (23) ρ f e,z ρ f e v+ q h FORMULATION FOR A PERFECT GAS Since the Navier-Stokes equations contain seven unknown field variables (ρ,u,v,w,e, p,t), two additional equations of thermodynamic relations between the state variables needs to be presented. In pure aerodynamics, it is generally reasonable to assume that the working fluid behaves like a calorically perfect gas, for which the equation of state takes the form [10]: p=ρrt, (24) where r denotes gas constant per unit mass, and is equal to the universal gas constant divided by molecular mass of the fluid. The internal energy e and the enthalpy h are only function of temperature: e=c v T = 1 p γ 1 ρ, h=c pt = γ p γ 1 ρ, (25) where c p and c v are the specific heat coefficients and γ is the ratio of specific heat coefficients under constant pressure and constant volume: γ= c p, c p = γ r. (26) c v γ 1 The pressure can be expressed as a function of ρ, E and the kinetic energy: p=(γ 1)ρ, [E u2 + v 2 + w 2 ]. (27) 2 The temperature can then be calculated with the use of relationship (24). The dynamic viscosity coefficient µ for perfect gas is strongly dependent on temperature but only weakly on pressure. Usually it is computed by the use of Sutherland s Law (5). 2.2 DIFFERENTIAL FORM The Equation (15) can be also written using the Gauss s theorem and introducing Φ i and Φ v. UdC+ ( Φ i Φ v )ds= Q s dc. (28) t C S C 9

25 Equation (28) can then be written for an arbitrary control volume C in the differential form: U t + ( Φ i Φ v )=Q s. (29) The generalized inviscid flux vector is: Φ i = E i i+f i j+ G i k, (30) where E i, F i and G i are respectively the x-, y- and z-components of Φ i, and are given by: ρu ρv ρw ρu 2 + p ρuv ρuw E i = ρuv, F i = ρv 2 + p, G i = ρvw. (31) ρuw ρvw ρw 2 + p ρuh ρvh ρwh The generalized viscous flux vector is: Φ v = E v i+f v j+ G v k, (32) where E v, F v and G v are respectively the x-, y- and z-components of Φ v, and are given by: τ xx τ yx τ zx E v = τ xy, F v = τ yy, G v = τ zy τ xz τ yz τ zz uτ xx + vτ xy + wτ xz q x uτ yx + vτ yy + wτ yz q y vτ zx + uτ zy + wτ zz q z, where the scalars q x, q y and q z are the Cartesian components of the heat flux vector q: q= k T, (33) where k is the thermal conductivity, and T is the static temperature. We can than rewrite scalars q x, q y and q z in the form: and so: q x = k T x, q y = k T y, q z = k T z, (34) Φ i = E i x + F i y + G i z, (35) 10

26 Φ v = E v x + F v y + G v z. (36) Using the Equations (35) and (36), the Equation (29) now becomes: U t + E i x + F i y + G i z = E v x + F v y + G v z + Q s. (37) 2.3 COSA In house built compressible NS solver COSA is currently a two-dimensional (2D) research CFD solver. By solving the NS equations internal and external viscous flows can be computed, which are a system of N pde nonlinear partial differential equations (PDE s). For 2D laminar flows N pde = 4 because the momentum equation has only two scalar components. Given a control volume C with boundary S, the Arbitrary Lagrangian-Eulerian integral form of the 2D time-dependent NS equation is: ( ) U dc + ( Φ i Φ v ) d S=0. (38) t C(t) S(t) The array U of conservative flow variables is defined as: U =[ρ ρu ρv ρe], (39) where ρ, u, v and E are respectively the flow density, the x and y component of the flow velocity vector v, and the total energy per unit mass. The definition of the total energy is defined in Equation (7), and is in 2D: E = e+(u 2 + v 2 )/2. The generalized inviscid flux vector is given by Φ i : Φ i = E i i+f i j v b U, (40) where E i and F i are respectively the x and y components of Φ i, and are given by: ρu ρv ρu E i = 2 + p ρuv, F i = ρuv ρv 2. (41) + p ρuh ρvh The vector v b is the velocity of the boundary S, and the flux term v b U is its contribution to the overall flux balance, which is nonzero only in the case of unsteady problems with moving boundaries. The symbol p denotes the static pressure and the symbol H denotes the total enthalpy per unit mass, defined in Equation (14). 11

27 The generalized viscous flux vector Φ v is: Φ v = E v i+f v j. (42) where E v and F v are respectively the x and y components of Φ v, and are given by: 0 0 τ E v = xx τ, F v = yx. (43) τ xy τ yy uτ xx + vτ xy q x uτ yx + vτ yy q y The scalars q x and q y are the Cartesian components of the heat flux vector, defined in Equation (33). The scalars τ xx, τ xy, τ yx and τ yy are the Cartesian components of the stress tensor τ, defined in Equation (3). Such tensor depends on the divergence of the flow velocity vector v, and the strain rate tensor, defined in Equation (4). The system of conservation laws represented by Equation (38) must be augmented with an equation of state. The perfect gas model has been used for all simulations reported in this diploma work. 12

28 3 GENERALIZED CURVILINEAR COORDINATES It is convenient to work with the curvilinear coordinate instead with the cartesian ones, in order to account for arbitrary body-fitted grids. In Appendix A [9] the relationships between the physical and the computational coordinates are shown and the physical features of the transformation are explained. For a two dimensional problem, we seek to relate points in(x,y) to corresponding points in (ξ,η) space. Therefore, the relationships ξ=ξ(x,y) and η=η(x,y) are required. Given such relationships, it is then possible to evaluate the equations of motion in (ξ, η) space. Applying the chain rule to the elements of the Equation (37) one can obtain: way: E i x = ξ E i x ξ + η E i x η = ξ x E iξ + η x E iη, (44) F i y = ξ F i y ξ + η F i y η = ξ y F iξ + η y F iη, (45) E v x = ξ E v x ξ + η E v x η = ξ x E vξ + η x E vη, (46) F v y = ξ F v y ξ + η F v y η = ξ y F vξ + η y F vη. (47) Substituting the above equations and dividing them by J: ( ) U + ξ x E iξ + η x E iη + ξ y F iξ + η y F iη = ξ x E vξ + η x E vη + ξ y F vξ + η y F vη Q + s J J J J J J J J J J. (48) t The terms in Equation (44) can be rearranged as follows: ( ) ( ) ξ x E iξ ξ x E i ξx = E i. (49) J J J ξ ξ Using the Equation (95) for the 2D case one can rewrite the Equation (49) in the following ξ x E iξ J = ( ξ x E i J ) ξ E i (y η ) ξ. (50) It is possible obtaining similar relationships for the other terms in Equation (48). With this changes and using the assumption: ξ η = η ξ, (51) we obtain: ( ) U + J t ( ) ξx E i +ξ y F i J + ξ ( ) ηx E i +η y F i J = η 13 ( ) ξx E v +ξ y F v J + ξ ( ) ηx E v +η y F v J η + Q s J. (52)

29 We can write: U = U J = J 1[ ρ ρu ρv ρe Φ i,1 = ξ x E i + ξ y F i J Φ i,2 = η x E i + η y F i J ρv c1 ], (53) = J 1 ρuv c1 + pξ x, (54) ρvv c1 + pξ y ρhv c1 ρv c2 = J 1 ρuv c2 + pη x, (55) ρvv c2 + pη y ρhv c2 where the contravariant velocities in the ξ, η,coordinate directions are given by: V c1 = uξ x + vξ y, V c2 = uη x + vη y. (56) (57) And for the viscous fluxes: 0 Φ v,1 = ξ x E v + ξ y F v = J 1 ξ x τ xx + ξ y τ xy J, (58) ξ x τ yx + ξ y τ yy ξ x Θ x + ξ y Θ y 0 Φ v,2 = η x E v + η y F v = J 1 η x τ xx + η y τ xy J, (59) η x τ yx + η y τ yy η x Θ x + η y Θ y and finally: Q s = Q s J = J 1 0 ρ f ex ρ f ey ρ f e v+ q h. (60) Substituting the Equation (52) one can obtain: U t + Φ i,1 ξ + Φ i,2 η = Φ v,1 ξ + Φ v,2 η + Q s. (61) 14

30 4 CFD SOLVER 4.1 SPACE DISCRETIZATION The cell-centered structured multi-block finite volume parallel CFD code COSA [5] solves the integral form of the time-domain conservation laws expressed by system (38) on structured grids in generalised curvilinear coordinates making use of a second order upwind spacediscretisation. The code can also solve a nonlinear frequency-domain formulation of the NS equations using a harmonic balance approach [6]. The parallelization of the time- and frequencydomain solvers is presented in [12]. The discretization of the convective fluxes is based on Van Leer s Monotone Upstreamcentered Schemes for Conservation Laws (MU SCL) extrapolations and the approximate Riemann solver of Roe s flux-difference splitting. The nonlinear system of equations is solved explicitly using a multi-stage Runge-Kutta (RK) smoother, with implicit residual smoothing. If vector n is the outward normal of the face of a grid cell, and ds is the area of such a face, the numerical approximation of the convective flux component Φ i, f =( Φ i n)ds through such face is: Φ i, f = 1 [ Φ i, f (U L )+Φ i, f (U R ) Φ i, f 2 U ]. δu (62) The superscript, the subscript f, and the subscripts L and R denote numerical approximation, face value, and value extrapolated from the left and from the right, respectively. The numerical dissipation depends on the generalized flux Jacobian Φ i, f / U and the flow state discontinuity across the cell face, defined by δu =(U R U L ). The discretization of the diffusive fluxes is based on second order centered finite-differences. The Cartesian derivatives of the flow velocity components are computed with the chain rule, using the derivatives of such components with respect to the local generalized curvilinear coordinates associated with the grid lines, and the grid metrics. 4.2 NUMERICAL INTEGRATION OF TIME-DEPENDENT EQUATIONS COSA solves the system of ordinary differential equation s: V dq dt + R Φ(Q)=0. The entries of the array Q are the unknown flow variables at the N cell cells discretizing the computational domain. The array Q can be viewed as made up of N cell sub-arrays, each of 15 (63)

31 which stores the N pde flow unknowns at a particular physical time. The length of Q is therefore (N pde N cell ). The array R Φ stores the cell residuals, and its structure is the same as that of Q. For each cell, the N pde residuals are obtained by adding the convective fluxes Φ i, f and the viscous fluxes Φ v, f through all the faces of the cell. The diagonal matrix V stores the volumes of the grid cells. It can be viewed as a block-diagonal matrix of size (N cell N cell ) with each block being the identity matrix of size (N pde N pde ) multiplied by the volume of the cell the block refers to. Implicit second order backward finite-difference is used for the discretisation of the physical time-derivative dq/dt of system (38), to march in physical time t: ( R g Q n+1 ) = 3Qn+1 4Q n + Q n 1 ( V + R Φ Q n+1 ) 2 t The symbol R g denotes a residual vector which also includes the source terms associated with the discretization of the physical time-derivative. Note that V is independent of the physical time-level (denoted by the superscripts n+1, n and n 1) because in this diploma work, only rigid-body grid motion is considered. The symbol t indicates the user-given physical time-step. Equation (64) can thus be viewed as a system of (N pde N cell ) ordinary differential equations in which the unknown is Q n+1, the flow state at time-level n+1. The set of nonlinear algebraic equations resulting from the space- and time-discretization of system (38) is then solved with an explicit scheme based on the use of a fictitious time-derivative (Jameson s dual-time-stepping [13]). The introduction of the pseudo time derivative τ, yield the equation: V dq n+1 + R g (Q n+1 )=0, (64) dτ The calculation of Q n+1 is performed iteratively by discretizing the fictitious time-derivative (dq n+1 /dτ) of Equation (64) with a four-stage RK scheme, and marching the equations in pseudo-time until a steady state is achieved. Such steady state is the flow solution for the physical time being considered. The convergence rate is then greatly enhanced by means of local time-stepping (LTS), variable-coefficient central implicit residual smoothing (IRS) and a full-approximation scheme multigrid (MG) algorithm. This solution procedure may become unstable when the physical time-step t scheme is significantly smaller than the pseudo-time-step τ. This instability was reported in [1], and thoroughly investigated by Melson et al. [20]. The latter study elegantly solved the stability problem by treating implicitly the Q n+1 term of the physical time-derivative within the RK 16

32 integration process. This strategy has also been implemented in COSA, as summarized below. The residual R g is split into the contribution depending on the Q n+1 term of the physical timederivative, and a term R d equal to the difference of R g and the aforesaid Q n+1 term: R g (Q n+1 )= V [ ] 3 t 2 Qn+1 + g(q n,q n 1 ) + R Φ (Q n+1 ), (65) where: g(q n,q n 1 )= 2Q n + 0.5Q n 1. This equation can also be written as: R g (Q n+1 )=R d (Q n+1 )+ 3V 2 t Qn+1. (66) In the previous work of Jameson [13], all contributions from the physical time term were carried to the right-hand side of the equation and treated as explicit terms within the RK stages: W 0 = Q n+1 l W k = W 0 α k τ V R g( W k 1 ), k=1,ns Q n+1 l+1 = W NS, where k denotes the RK stage index and it varies between 1 and the number of RK stages NS, α k is the k th RK coefficient, l is the RK cycle counter. In many flows, especially high Reynolds number viscous flows, coefficient β = 1.5 τ/ t may easily become large and the scheme can become unstable. COSA uses modified algorithm, which is stable for all values of β, by treating the contribution of the physical time derivative implicitly within the Runge-Kutta scheme as previously mentioned. Discretizing the fictitious time-derivative of Equation (64) with a multi-stage RK scheme, introducing the decomposition of R g provided by Equation (66), and considering the Q n+1 term at stage k rather than at stage (k 1) yields the following RK algorithm: W 0 = Q n+1 l (1+α k β)w k = W 0 α k τv 1 R d (W k 1 ) (67) Q n+1 l+1 = W NS. The stability analysis of [20] shows that the stability of algorithm (67) no longer depends on the ratio τ/ t. However this formulation is still unsuitable when IRS and MG are also used, because both acceleration techniques have to be applied to a residual term that vanishes at 17

33 convergence, and this is not the case of R d. The solution is to introduce the residual R g which does vanish at convergence. Given that: τr d (W)= βvw + τr g (W), the IRS-MG-tailored counterpart of algorithm (67) is: W 0 = Q n+1 l (1+α k β)w k = W 0 + α k βw k 1 α k τv 1 L IRS [R g (W k 1 )+ f MG ] Q n+1 l+1 = W NS, (68) where L IRS denotes the IRS operator, and f MG is the MG forcing function, which is nonzero when the smoother (68) is used on a coarse level after a restriction step. Note that the matrix multiplying W k at the second line of algorithm (68) is diagonal, and this implies that for each grid cell the N pde unknowns can be updated without an actual matrix inversion. Algorithm (68) is based on a point-implicit Runge-Kutta (PIRK) integration of the time-dependent NS equations. The standard fully-explicit Runge-Kutta (FERK) integration method is retrieved by setting β = 0 in this algorithm. The integration scheme of the steady equations is instead obtained by also replacing R g with R Φ in algorithm (68). 18

34 5 LOW-SPEED PRECONDITIONING In the case of inviscid steady low-speed flows, a large disparity between the convective and acoustic eigenvalues of the flux Jacobian Φ i, f / U exists. This results in unbalanced amounts of numerical dissipation, and this occurrence spoils the accuracy of the solution. The basic idea of preconditioning is to pre-multiply the time derivative by a preconditioning matrix in order to control eigenvalues of the flux Jacobian. In physical terms, preconditioning tries to rescale the rapidly propagating acoustic speeds to a value comparable to particle velocity. The matrix premultiplication applies only to the time-derivative term and does not affect the steady state solutions. When using explicit time-marching methods, the local time-step also depends on the eigenvalues of the flux Jacobian, and a large disparity between convective and acoustic speeds substantially impairs the convergence rate of the solver. An analogous disparity among the eigenvalues of the Jacobian of the governing equations also occurs in the case of viscous steady and time-dependent low-speed problems. This issues are solved by using low-speed preconditioning, which changes the eigenvalues of the system of compressible flow equations in order to remove large disparity of wave speeds. Premultiplied time derivatives by a matrix which slows down the speed of acoustic waves result, that all the waves at low Mach numbers will have the same speed and the system becomes well-conditioned [25, 26]. When dealing with steady low-speed flows, the superscript (n+1) in Equation (64) can be dropped since there is a single unknown state corresponding to the sought steady solution, the residual vector R g of Equation (64) is replaced by the residual vector R Φ, and the continuous counterpart of the pseudo-time derivative dq dτ is premultiplied by a preconditioning matrix (Γ cs ) 1. This results in a rescaling of the eigenvalues of the preconditioned sum of the convective and viscous flux Jacobians which restores the correct levels of numerical dissipation. This allows that high convergence rates are maintained even with low-speed problems. The preconditioner Γ cs used by COSA is that proposed in [26], where its expression can be found. The matrix Γ cs depends on a parameter M ps. The choice M ps = 1 yields no preconditioning. For low-speed flows, the parameter M ps is: M ps = min(max(m,m pg,m vis,ε),1), (69) where M is the actual local Mach number, M pg is a cut-off value based on the local pressure gradient [28, 8], and M vis is the viscous cut-off value and is used when resolving the NS equa- 19

35 tions (i.e. viscous flows). It is based on the cell Reynolds number, also called Peclet number and is defined by: M vis = max M 2 ξ ( 1 Re ξ 1) Re ξ [1+M 2 ξ ( 1 Re ξ 1)], M 2 η ( 1 Re η 1) Re η [1+M 2 η( 1, (70) Re η 1)] where M ξ and M η are respectively the Mach number based on the ξ and η contravariant velocity components, and Re ξ is the Reynolds number based on the ξ contravariant velocity component and the cell length in the ξ direction, and Re η is the counterpart of Re ξ for the η direction. The derivation of Equation (70) can be found in [4]. The parameter ε is a small cut-off parameter that prevents the preconditioner from becoming singular at stagnation points. The introduction of LSP modifies the artificial dissipation term of the numerical flux provided by Equation (62) as follows: Φ i, f = 1 [ Φ i, f (U L )+Φ i, f (U R ) Γ 1 cs 2 Γ Φ i, f cs U ]. δu (71) For steady problems, the use of Equation (69) to build Γ cs and its inverse guarantees both the balance of the numerical dissipation and an optimal convergence rate. For time-dependent problems, the report [26] has proposed a preconditioning matrix Γ cu with an identical structure to that of Γ cs, but depending on an unsteady preconditioning parameter defined as: M pu = min(max(m,m pg,m vis,m uns,ε),1), (72) where M uns is a cut-off value based on the physical time-step t and the characteristic lengths of the domain l [26], it is defined as: M uns = l πc s t, (73) where c s is the local sound speed, l = max(l x,l y ) and l x and l y are the characteristic dimensions in the x and y directions respectively. The use of Equation (72) to calculate the entries of Γ cu usually yields significantly higher convergence rates with respect to those observed using Γ cs, but the use of Γ cu does not guarantee an optimal scaling of the artificial dissipation. This has been observed by Pandya et al. [21] and Housman et al. [11] for time-dependent problems with motionless grids, and by Campobasso et al. [7] also for the case of time-dependent problems with moving grids. The article [7] also presented a mixed preconditioning strategy to overcome this problem, and demonstrated its 20

36 effectiveness with a number of time-dependent problems with motionless and moving grids. With mixed preconditioning, the convective numerical flux is: Φ i, f = 1 [ Φ i, f (U L )+Φ i, f (U R ) Γ 1 cu 2 Γ Φ i, f cs U ]. δu (74) In the CFD code COSA, the cell residuals obtained by imposing the flux balance of the convective fluxes of Equation (74) and the viscous fluxes are premultiplied by the preconditioner Γ cu before updating the solution on the current grid level. As a result, the steady preconditioning matrix Γ cs is used to build the preconditioned numerical dissipation. Conversely, the local time-step used to advance the solution is computed using the eigenvalues of the preconditioned flux Jacobian Γ cu Φ i, f / U. When using the preconditioning, also the explicit integration process based on the implicit treatment of the RK scheme and the IRS operation is modified. Equation (64) now becomes: V(Γ 1 U cu τ )n+1 + R g (U n+1 )=0. (75) The general form of the standard RK-IRS-MG iteration to solve low-speed time-dependent problems, obtained by premultiplying the fictitious time-derivative of Equation (64) by Γ 1 cu, modifying the numerical dissipation as indicated by Equation (74), and discretizing this derivative with the multistage RK of choice, is: W 0 = Q n+1 l W k = W 0 α k τv 1 L IRS Γ cu [R g (W k 1 )+ f MG ] Q n+1 l+1 = W NS. The use of the stabilization process of the RK cycle discussed in the previous subsection yields the following stabilized PIRK iteration: (76) W 0 = Q n+1 l (I+ α k βγ k 1 cu )W k = W 0 + α k βγ k 1 cu W k 1 α k τv 1 L IRS Γ k 1 cu [R g (W k 1 )+ f MG ] Q n+1 l+1 = W NS. (77) The matrix premultiplying W k is block-diagonal, but its blocks are not diagonal because of the preconditioner Γ cu, which is not a diagonal operator. Therefore the update process requires the inversion of an (N pde N pde )-matrix for each cell of the computational domain. The FERK integration algorithm (76) of the time-dependent equations for low-speed problems is retrieved by setting β = 0 in this algorithm. The integration scheme of the steady equations with LSP is instead obtained by also replacing R g with R Φ and setting M uns = 0 in algorithm (77). 21

37 6 OSCILLATING WING DEVICE An oscillating wing device is defined as an airfoil experiencing simultaneous pitching θ(t) and heaving h(t) motions. This promising device could be used in the field of both wind and water power production. In case of using it for water applications, it appears to be environmentally more acceptable than conventional hydropowerplants. It could be used in slow-flowing rivers where dams are not practical due to ship traffic and low terrain. Due to the low velocities it could also be used for tidal energy-production. It the field of wind turbines it could be used for small-scale power production instead of vertical- and horizontal-axis wind turbines. The following mathematical representation of the imposed motion is that adopted in [16]. Taking a pitching axis located on the chord line at position x p from the leading edge (LE), the airfoil motion is expressed as: θ(t)=θ 0 sin(ωt) Ω(t)= θ 0 ωcos(ωt), (78) h(t)=h 0 sin(ωt+ φ) v y (t)= h 0 ωcos(ωt+ φ), (79) where θ 0 and h 0 are respectively the pitching and heaving amplitudes, Ω is the pitching velocity, v y is the heaving velocity, ω is the angular frequency and φ is the phase between the two motions. In this study, φ is set to 90 o, and the symmetric NACA0015 airfoil is considered. The freestream velocity is denoted by u and the angular frequency ω is linked to the vibration frequency f by the relationship ω=2π f. The prescribed oscillating motion is depicted in the Figure 1. y h 0 t /T =0 θ 0 t /T =1 x t / =0.25 T t /T =0.5 t /T =0.75 Figure 1: Prescribed motion of an oscillating wing. The arbitrary phase angle between the heaving and pitching motions φ determines how much work is done on the airfoil and consequently how much energy is extracted from the flow. The effect of phase angles of 90 and 0 degrees is illustrated in Figure 2, viewed in the freestream velocity reference frame (apparent motion of the airfoil is from right to left). Dashed arrows 22

38 v y C Y t /T =0.25 t /T =0 t /T = Figure 2: Effect of phase angle on power-extraction. represent the direction of heaving velocity v y and the solid arrows show the direction of heaving force coefficient C Y. The top sketch depicts the case where pitching and heaving are 90 degrees out of phase, where pitching leads. Bottom sketch represents the phase angle zero between pitching and heaving. Since the heaving force coefficient and the heaving velocity always point in the same direction, as it is evident on the top sketch, work is positive all the time. If they are pointing in the same direction during part of the cycle, they are producing positive work and if they are in anti-phase during the other part of the cycle, they are producing negative work. The net work produced throughout a cycle with zero phase difference is zero, because the positive and negative work cancel out. The phase angle between pitching and heaving motions is one of the critical parameters which may determine whether the airfoil extracts the power from an oncoming flow. Another key parameter of an oscillating symmetric airfoil is the effective angle of attack (AoA) α. It can operate in two different regimes, namely, in propulsive or power-extracting mode. This distinction originates from the sign of the forces that the flow generates on the oscillating airfoil. The effective AoA varies at different points on the airfoil surface and is dependent on the pitching axis location. Based on the imposed motion and the upstream flow 23

39 conditions, the airfoil experiences an effective AoA α and an effective velocity v e given by: α(t)=arctan( v y (t)/u ) θ(t), (80) v e (t)= u 2 + v y(t) 2. (81) The maximum values of α and v e have a major impact on the amplitude of the peak forces in the cycle, and also on the occurrence of dynamic stall. The maximum effective angle of attack reached in the cycle is approximated by the modulus of its quater-period value: ( ) ωh0 α max α(t/4) = arctan θ 0. (82) u The maximum effective velocity also occurs at the quarter-period: (v e ) max v e (T/4) = u 2 +(ωh 0) 2. (83) As explained in [16] and shown in the Figure 3-a, the power-extracting regime is achieved when α(t/4) < 0. Three sketches in Figure 3 present a time-sequence viewed in a reference a. power extraction y r x r t /T =0.25 t/t =0 α Y R L R D t /T =0.5 X b. feathering c. propulsion Figure 3: Effect of pitch amplitude on power-extraction. 24

40 frame moving with the farfield flow at u, so that the effective angle of attack α(t) is made visible from the apparent trajectory of the airfoil. In the sketch a, the resultant force R is first constructed from typical lift and drag forces, on right-hand side, and then decomposed into X and Y components on left-hand side. One can see that the resultant aerodynamic force R has a vertical component Y that points in the same direction as the vertical velocity component v y of the airfoil, and this implies that the wing extracts energy from the fluid as long as no energy transfer associated with the component X of the aerodynamic force takes place. This is clearly the case since the airfoil does not move horizontally. As shown in sketch b, if α(t/4) 0 the airfoil is feather through the flow. The last sketch c of Figure 3 shows the case where α(t/4)>0 and in this case an oscillating wing device generates thrust. The aerodynamic analyses reported in [16] and those reported in the result section of this diploma work highlight that the aerodynamic phenomena taking place during the wing oscillation are significantly more complex than the quasi-steady model discussed above. More specifically, the extent and efficiency of the energy extraction are heavily influenced by the occurrence of unsteady leading edge vortex shedding (LEVS) associated with dynamic stall and the LEVS timing with respect to the airfoil motion. Taking a wing of span equal to one unit length, the instantaneous power extracted from the flow is the sum of a heaving contribution P y (t) = Y(t)v y (t) and a pitching contribution P θ (t) = M(t)Ω(t), where M is the resulting torque about the pitching center x p. Denoting by c the airfoil chord, and C P P/( 1 2 ρ u 3 c) a power coefficient, the nondimensional power extracted over one cycle is given by: C P = C Py +C Pθ = 1 T [ C Y (t) v y(t) +C M (t) Ω(t)c ] dt, (84) T 0 u u where the heaving force coefficient is given by the relationship: C Y (t)= Y(t) ( 1 2 ρ u 2 c), and the torque coefficient is written as: C M (t)= M(t) ( 1 2 ρ u 2 c2 ). The efficiency η of the power extraction is defined as the ratio of the extracted mean total power P and the total available power P a of the oncoming flow passing through the swept area or in other words the flow window: η P = P y+ P θ P 1 a 2 ρ u 3 d = C c P d, (87) 25 (85) (86)

41 where d is the overall vertical extent of the airfoil motion. This distance considers both heaving and pitching motions, and is typically slightly larger than 2h 0. The upper limit of the efficiency η is 59 %, and this result follows from Betz s analysis [2] of a stationary inviscid streamtube around a power-extracting device. 26

42 7 VALIDATION Validation is an essential step to verify the accuracy of the code. For validation one must separately demonstrate that the algorithms employed in the code are suitable approximations of the physics. To validate the code one can use either the numerical solution obtained by another code or an analytical solution of governing equations. Using the result of another code to validate the accuracy of the code in question assumes that the other code has been successfully verified, which adds a certain amount of subjectivity into what should be strictly objective mathematical exercise. Using the analytic solutions to the governing equations is a much more reliable method. The second order accuracy of the convective flux discretization has been verified by computing the solution of a 2D inviscid steady problem for which the analytical solution has been determined. While solving this problem, several grids have been used, which become successively finer by a factor of two in both directions. Analyses of the root mean square (RMS) of the error between the analytical solution and the computed solutions obtained by using these grids have confirmed the second order of the space-discretization [5]. The second order accuracy of the time- and space-discretization of the solver using LSP has been demonstrated by considering an unsteady test case resulting from the superposition of a uniform low-speed flow and a steady vortex. The analytical solution of the Euler equations for this problem has been used to highlight the second order accuracy of COSA for this type of problem [7]. The predictive capability of the code for inviscid problems is verified in [6], which compares the time-dependent pressure difference across a pitching flat plate in a uniform freestream to the analytical solution of Theodorsen [24]. Here, the predictive capabilities of the steady and time-dependent viscous solvers with LSP are assessed by computing a laminar boundary layer over a flat plate, and the flow past a circular cylinder shedding Von Karman vortices. 7.1 LAMINAR FLAT PLATE The viscous laminar flow over a flat plate leading to the formation of a laminar boundary layer is considered. The computational domain is rectangular and the flat plate lies on the lower horizontal boundary. The LE of the flat plate is in the origin of the Cartesian system, and its trailing edge (TE) is at x = 1, where the outlet boundary is vertically positioned. The inlet boundary is at x = 0.25, and the upper horizontal side is a farfield boundary positioned at y = 27

43 4. The uniform Cartesian grid has 1025 points along the y-axis and 321 points along the x- axis. 256 points lie on the flat plate and 64 in the space between the LE and the inlet boundary. The freestream Reynolds number is set to From a physical point of view, the effects of compressibility are expected to be negligible for M of order 0.1 or less. When using the compressible formulation without LSP, however, both the convergence rate of explicit solvers and the accuracy of the solution, are expected to worsen as the Mach number reduces. In order to assess the effectiveness of the developed LSP technique, this problem has been solved for three values of M, namely, 0.1, 0.01 and 0.001, and for each value a simulation with LSP and one without LSP have been performed. All solutions have been compared with Blasius self-similar solution of the incompressible boundary layer equations [22]. C f M 0.1 M 0.01 M Blasius VP x 0.04 NP x Figure 4: Comparison of skin friction coefficient determined with Blasius solution and computed with CFD simulations. The skin-friction coefficient C f on the flat plate computed with the six CFD simulations is reported in Figure 4. The left plot labeled VP (viscous preconditioning) refers to the calculations using LSP with preconditioning parameter defined by Equation (69), whereas the right plot labeled NP (no preconditioning) refers to simulations without LSP. Both plots also report the theoretical solution of Blasius. One can see that the three solutions in left plot with LSP are indistinguishable and in a very good agreement with the Blasius solution. However, the solutions without LSP in right plot are not all superimposed, as that with M = presents strong oscillations at the LE, and also a significantly different level towards the TE. Figure 5 represents the comparison of the x and y velocity components at x 0 = 0.5, obtained with the six CFD simulations, with the theoretical solution of Blasius. The top plots are for the x 28

44 1.2 VP 1.2 NP u/u M M 0.01 M Blasius VP 0.05 NP v/u η w η w Figure 5: Comparison of velocity profiles determined with Blasius solution and computed with CFD simulations. velocity component and the bottom plots are for y velocity component, which is nondimensionalized by the freestream velocity against the nondimensional wall distance η w = y u/( x 0 ), in which denotes the kinematic viscosity. Left plots labeled VP refers to the three simulations with LSP and the right plots labeled NP refers to the three simulations without LSP. The left plots show that the three CFD solutions associated with the three selected M are equal, as expected on the basis of physical evidence, and in very good agreement with the solution of Blasius. The difference between the computed and the theoretical profiles of the y velocity component in bottom left plot is due to the fact that the farfield boundary conditions of the CFD simulation and the theoretical solution are different and applied at different locations. The former is applied 4 chords above the flat plate, and the latter at the edge of the boundary layer. The right plots show that the CFD solutions without LSP are not independent of the Mach number, as the solution associated with M = differs both from the other two CFD results and the solution of Blasis. The results of Figures 4 and 5 provide evidence of the necessity 29

45 of using LSP to preserve the accuracy of the solution when solving low-speed flows with the compressible flow equations l r -12 M 0.1, NP M 0.1, VP M 0.01, NP M 0.01, VP M 0.001, NP M 0.001, VP ρ ρu l r ρv MG cycle ρe MG cycle Figure 6: Convergence histories of CFD simulations. Figure 6 highlights the impressive enhancement of the computational efficiency of the explicit multigrid compressible solver enabled by LSP for low-speed problems. Four plots report the convergence histories of conservation equations, namely, the top left plot labeled ρ is for the continuity equation, the top right and the bottom left are for the x component of the momentum equation labeled ρu and the y component of the momentum equation labeled as ρv. The bottom right plot is for the energy equation labeled ρe. In all plots, the variable on the x-axis is the number of multigrid iterations, and the variable l r on the y-axis is the logarithm with base 10 of the RMS of all cell-residuals for the considered conservation equation. Each plot reports the convergence history of the CFD runs with and without LSP for the three selected values of M. The LSP analyses have been run until the maximum value of l r among the four equations was smaller than 1.d 14. This condition was achieved between about 250 and 350 MG cycles for all three values of M when using LSP. One can see that, for a given equation, the convergence rate of all three LSP simulations is independent of M, as expected on the 30

46 basis of theoretical analyses. It should also be noted that the convergence rate of the LSP simulations using inviscid preconditioning, namely setting M vis = 0 in the definition of the steady preconditioning parameter M ps defined by Equation (69) is substantially lower than that achieved by using the present viscous preconditioning [27]. The use of either methods does not change the accuracy of the solution, and this highlights the benefits of optimizing the design of the LSP methodology. The simulations without LSP have been run until either the maximum value of l r among the four equations was smaller than 1.d 14 for all four equations or 5000 MG cycles were performed. The simulation for M = 0.1 has required 620 MG cycles, that for M = MG cycles, and that for M = has terminated after 5000 MG cycles, as the requested minimum residual could not be achieved. One notes that the convergence rate of the three simulations is not independent of M, and it dramatically worsens as this parameter is reduced. The solutions for M = reported in Figures 4 and 5 refer to the state obtained with 5000 MG iterations. Performing about 5000 additional MG cycles reduces the errors of this solution with respect to that obtained with LSP. This latter, however, is obtained with little more than 200 MG iterations, and this emphasizes the computational benefits of LSP. 7.2 VORTEX SHEDDING OF CIRCULAR CYLINDER The accuracy of the COSA solver with LSP for laminar unsteady flows with motionless grid is assessed by considering the vortex shedding behind a circular cylinder. The unsteady flow generates periodic shedding cycle known as Von Karman Street, caused by the unsteady separation of a fluid flow over the cylinder. As the wake behind cylinder starts to grow, it becomes unstable and begins to shed vortices from alternate sides of the cylinder. The Reynolds number based on the cylinder diameter d c equals 1200 and the freestream Mach number is The O-type computational grid was used for the analysis and it has 193 points in the circumferential and 93 points in the radial direction. The farfield boundary is at 20 cylinder diameters from the center of the cylinder, and the minimum wall distance is 0.1 % of the diameter. The adopted grid has a number of nodes similar to that of the unstructured grid made up of triangular cells used in [15] to simulate the vortex shedding behind the cylinder for M = 0.2, and the minimum wall distance and the distance of the outer boundary from the cylinder are also the same in both grids. The vortex shedding simulation has been performed using nine values of the nondimensionalized physical time-step t. Starting from t 0 = 400, the other 8 values 31

47 Table 1: Input and output parameters of nine LSP simulations of vortex shedding. t i St N T CFL Alg FERK FERK PIRK PIRK PIRK PIRK PIRK PIRK PIRK are given by: t i = t i 1 /2, i=1,8. One of the output variables of the CFD analysis is a measure for the frequency f sh of the vortex shedding, typically quantified by means of the Strouhal number St = f sh d c /u. The Strouhal number obtained with the nine CFD simulations using mixed-preconditioning is reported in the second column of Table 1, the first column of which provides the corresponding t i. The variable N T in the third column is the approximate number of time-intervals per period resulting from the particular choice of t i and the resulting St. Variations of St below 0.2 % are only obtained by using physical time-steps yielding 168 timeintervals per period. The result St differs by about 3.5 % from that reported in [15] for the analysis using the same t. The grid used in that analysis appears to have been obtained by splitting each quadrilateral of an structured O-type grid with circumferential spacing decreasing from the front to the back of the cylinder to better resolve the vortex trail. Since that grid has higher node density behind the cylinder with respect to that used here, and that, for a given overall number of grid nodes the unstructured grid obtained as described above has twice the cells of the parent structured grid, this agreement is very good. The CFL number used for the nine analyses are reported in the fourth column. The acronym in the fifth column indicates whether the FERK or PIRK MG integration has been used. For the reported CFD values, only the first two simulations could be performed by using the FERK MG integration. All others could be performed only by using the stabilized MG integration 68. Figure 7 shows the lift, drag and pitching moment coefficients obtained with the nine se- 32

48 2 1 C L l 0-1 nt 13 nt 23 nt 43 nt 85 nt 168 nt 336 nt 671 nt 1341 nt C D C M t N T 13 N T 23 N T 43 N T 85 N T 168 N T 336 N T 671 N T 1341 N T 2683 Figure 7: Analysis of time-resolution of forces acting on vortex shedding. lected values of t. All nine simulations have been run for an overall duration of nondimensional time-units, and have all been started from the same state, namely a point of the shedding cycle determined after periodicity of the solution with t = 25 had been achieved. These curves show that negligible variations of the forces occur once at least 168 intervals per period ( t = 25) are used, since the curves for N T 168 are indistinguishable. The second order accuracy of the LSP PIRK MG integration has been assessed by means of Richardson s extrapolations. Denoting by F i a generic local or integral output variable of the CFD simulation carried out with physical time-step t i, a second order time-discretization implies that F i = F exact + c 1 ti 2 where F exact is the unknown exact value of the considered 33

49 10 2 x C L δ i x x x x x x C D C M 2 nd ord. x x x x x x x x x x x x x x x x x x x 10-6 x t i Figure 8: Analysis of the order accuracy by means of Richardson s extrapolations based on the results of the time-refinement analysis of the vortex shedding cylinder. output and c 1 is a constant. Using this equation, it readily follows that: δ i = log F i F i 1 2log t i (88) This expression has been evaluated by assigning to F i, in turn, the values of the lift, drag, and pitching moment coefficients at t = computed with the nine selected t i s, and corresponding to the right end of the curves of Figure 7. The result of this operation is reported with three solid lines in Figure 8, which also provides three dashed lines of slope 2. This analysis confirms the correctness of the implementation, as the slope of all three computed curves becomes the theoretical one when t becomes suitably small. 34

50 8 RESULTS In this section the 2D laminar flow analyses of an oscillating wing simultaneously heaving and pitching are presented. It is subdivided into two subsections, namely, the first subsection consist of thorough investigation of the unsteady flow mechanisms enabling the efficiency of the energy extraction to be controlled and possibly maximized. The second subsection assesses the accuracy and the computational performance of the COSA compressible NS solver enhanced with LSP to deal with low-speed effects. 8.1 AERODYNAMIC ANALYSIS The wing section selected for this study is the NACA0015 airfoil. Two operating regimes are considered, one is characterized by a high efficiency of the energy extraction, onward referred as case A, and the other is characterized by a lower efficiency, denoted as case B. The CFD simulations of both regimes have been performed with M = This value is much lower than that at which this device is expected to be used, and it has been selected to set a challenging condition for the COSA compressible solver with LSP. The Reynolds number based on the freestream velocity and the airfoil chord is set to 1100, the heaving amplitude h 0 equals one chord and the pitching center is at x p = 1/3. Case A is characterized by a pitching amplitude θ 0 of o and a nondimensionalized frequency f = f c/u of 0.14, where f is the frequency in Hertz. In case B, θ 0 and f are set to o and 0.18, respectively. The C-grid in Figure 9, used for all simulations, has 321 points along the airfoil, 353 points in the grid cut, and 257 points in the normal-like direction, giving the total number of cells. The farfield boundary is at about 50 chords from the airfoil, and the distance of the first grid points off the airfoil surface from the airfoil surface is about 0.02 % of the chord. In the time-dependent simulations, the whole grid is animated simultaneously by a heaving and a pitching motion component defined by Equations (78) and (79), respectively. The analysis of both operating regimes has been performed with the LSP-enhanced solver using 128 timeintervals per period and running the simulation for 8 cycles of oscillation. This has resulted in the maximum difference between C Y over the last two oscillation cycles being about 0.1 % of the maximum C Y over the last cycle. In both cases, the CFL number has been set to 3, and 1000 MG iterations per physical time-step have been performed. Using this configuration we observed a minimum drop of the RMS of all 4 residuals 4 orders of magnitude, and a 35

51 Figure 9: Two views of C-grid of NACA0015 airfoil. maximum drop of 7 orders. The simulations based on the aforementioned values of CFL and number of time-intervals per period could be performed only by using the stabilized PIRK MG iteration (77), as these calculations were numerically unstable otherwise C L C D h θ α C L, C D, h 0 0 θ, α C L, C D, h 0 0 θ, α C L C D h θ α t/t t/t a) b) Figure 10: Lift and drag coefficients of oscillating wing and kinematic variables over one period: a) θ 0 = o, f = 0.14 (case A), b) θ 0 = 60 o, f = 0.18 (case B). 36

52 In Figure 10, the lift and drag coefficients over the oscillation period for cases A and B are depicted. For both cases, these Figures also report the pitch angle θ, the vertical position h and the effective AoA α computed with Equation (80). The maximum AoA achieved in case A is about 35 o, whereas in case B is about 11.5 o. The C D curve of case A presents substantially higher values than those observed in case B, and this is partly due to higher boundary layer losses associated with the higher AoA and also to the higher pressure drag caused by higher level of flow separation. However, the C D curve of case A is also substantially higher than that of case B, and, unlike the latter case, it keeps the same sign of α until mid-period, achieving a second maximum at about 43 % of the period. As highlighted below, this phenomenon has a strong impact on the efficiency of the energy extraction process. The overall power coefficient C P, the power coefficient C Pθ associated with the pitching moment M, and the heaving force coefficient C Y for case A and case B are reported in Figure 11. Both figures also report the nondimensionalized heaving velocity v y /u. One observes that the peak value of C P is higher for case A, but more importantly the C P curve of case A remains positive over most of the cycle, whereas that of case B is negative over nearly 40 % of the cycle. As a result, the mean power coefficient for case A is C P = , whereas that for case B is C P = The values of the power coefficient C Pa associated with the available power in C Pa C Pa C P C Pθ C Y v y /u -2 C P C Pθ C Y v y /u t/t t/t a) b) Figure 11: Power and heaving force coefficients of oscillating wing and heaving velocity over one period: a) θ 0 = o, f = 0.14 (case A), b) θ 0 = 60 o, f = 0.18 (case B). 37

53 either case are also reported in Figure 11 for reference. The efficiency of the power extraction process computed by means of Equation (87) for cases A and B is respectively % and %. As highlighted in Figure 11 for case A, the reason why the C P curve of case A is positive over most of the cycle is, that the heaving force and heaving velocity keep the same sign over most of the period, whereas in case B they are in anti-phase between approximately 30 and 50 % of the cycle and between approximately 70 and 100 % of the cycle. The higher performance of case A thus arise from the fact that C Y remains higher in the second part of the semi-period for case A than it does for case B. One can also see the second maximum of C P for case A at 43 % of semi period, whereas in case B this does not occur. The reason for such a difference is the leading edge vortex shedding associated with the occurrence of dynamic stall in case A. A more detailed comparative analysis of the two regimes is presented in Figure 12, which refers to cases A and B, respectively. In both figures, the top and bottom left subplots, respeccase A case B Figure 12: Flow snapshots of case A and case B. Left plots are for vorticity (Ω f ) conturs at 25 % of the period (position 1) and at 45 % of the period (position 2). Right plots are for static pressure coefficient (C p ) contours for position 1 and 2. Dashed line contours denote positive C p and solid line contours denote negative C p. 38

54 tively, report the contours of the flow vorticity Ω f when the airfoil is in the positions 1 and 2 indicated in Figure 11. Position 1 corresponds to 25 % of the cycle (physical time index 33 out of 129), and position 2 corresponds to % of the cycle (physical time index 59 out of 129). The top and bottom right subplots provide instead the computed static pressure coefficient C p =(p p )/( 2 1ρ u 2 ) when the airfoil is in the two aforementioned positions. Dashed contour lines denote positive C p values, and solid contour lines denote negative C p values. The left top and bottom plots of Figure 12 for case A present a snapshot of Ω f corresponding respectively to the formation of the LE vortex (position 1) and the time at which such a vortex leaves the airfoil after traveling from its LE to its TE (position 2). Consequently, there exists a lowpressure region associated with the high-kinetic energy vortex, that also travels from the LE to the TE. The effect of such a moving low-pressure region is that of increasing the heaving force in the final portion of each semi-period and also making such a force remain longer in phase with the heaving velocity. This effect is highlighted by the C p contours of the right bottom plot of Figure 12, which shows that shortly before completion of the semi-period, the y-component of the net pressure force is still pointing downwards and therefore is still in phase with the heaving velocity, yielding a positive heaving power contribution. The LEVS just highlighted is due to the occurrence of dynamic stall and accounts for the two secondary peaks of C Y before completion of the two semi-periods visible in Figure 11-a, and the two secondary peaks of C L visible in Figure 10-a. As shown in Figure 12, no dynamic stall-induced LEVS is observed in case B. This explains the lower efficiency of the power extraction process for this regime and points to the importance of the LEVS and its synchronization with the airfoil motion for achieving high efficiency of the power conversion. Figure 11-a shows that about 50 % of the peak of the overall power coefficient C P in the second half of each semi-period is due to the pitching motion component C Pθ. This points to the impact that the selection of the pitching center x p is likely to have on the level on the extracted power and the efficiency of the process. Some parametric studies of this aspect are provided in [16], but a more extensive investigation on the effects of this parameter may lead to further improvements of the oscillating wing device. Finally, it should be noted that part of the graphical output presented in this subsection has been formatted in the same manner as its counterpart in the article of Kinsey and Dumas to enable one to compare the output of the computational framework of this paper with that of the FLUENT incompressible solver used in [16]. Despite the fact that the spatial and temporal resolutions of the two studies differ 39

55 Figure 13: Flow snapshots of high efficiency power-extraction case A. Vorticity (Ω f ) conturs and static pressure coefficient (C p ) contours are shown on left and right plots, respectively. significantly, their detailed results are in very good agreement. 40

56 To show more clearly when the LEVS starts to appear during each cycle, flow snapshots of the second part of semi-period are presented in Figure 13 for the high efficiency of the energy extraction case A. The left plots show the contours of the flow vorticity Ω f while on the right side, for the same process, the computed static pressure coefficient C p contours are shown. The top left snapshot shows the early formation of LE vortex. This phenomenon is confirmed with low pressure region on the right top snapshot. While the oscillating wing is finishing each semiperiod the vortex and low-pressure region are traveling from LE to TE. Just before the end of semi-period the vortex leaves the airfoil and the pressure around the airfoil is normalized. This state perseveres until the beginning of a new second part of semi-period. As explained before, this mechanism is particularly important to maximize the power-extraction efficiency, due to vortex formation. 8.2 PERFORMANCE OF LOW-SPEED PRECONDITIONING The flow analyses presented in the preceding subsection are based on simulations performed with M = using LSP. In order to assess the effectiveness of LSP in terms of both, solution accuracy and computational performance for this complex problem, the flow field associated with regime A at M = has also been computed without LSP. Additionally, simulations of the same operating regime have also been performed with and without LSP for M = 0.01 and M = 0.1. All six simulations have been run for 8 periods using 128 time-intervals per period. In all cases, the maximum difference between C Y over the 7 th and 8 th cycles is smaller than 0.25 % of the maximum C Y over the 8 th cycle. In all calculations with LSP the CFL number has been set to 3; this parameter has been set to 2 in all calculations without LSP, as the numerical stability of the integration could not be preserved using higher values. The number of MG iterations per physical time-step has not been the same for all simulations, but in all cases its choice has led to a minimum drop of the RMS of all 4 residuals of 4 orders of magnitude. In the case of the three simulations without LSP, the solution of one or two physical times per period has not achieved full convergence, due to premature stagnation of the residuals. This condition could not be removed by further lowering the CFL number. Fortunately, this occurrence is believed not to have any significant effect on the analysis presented in this subsection, since the stagnation of the residuals has been due to the lack of complete convergence only in small and localized areas of the flow field. For Mach numbers lower than 0.3, the effects of compressibility are negligible; therefore, 41

57 5 4 C X C Y C θ t/t M 0.1, MP M 0.01, MP M 0.001, MP M 0.1, NP M 0.01, NP M 0.001, NP Figure 14: Comparison of force coefficients of oscillating wing for three values of M obtained with and without LSP. Top plot: horizontal force coefficient. Middle plot: vertical (heaving) force coefficient. Bottom plot: Pitching moment coefficient. compressible flow analyses performed by imposing M 0.1 should yield the same solution in terms of nondimensional coefficients. The horizontal force coefficient C X, the vertical force coefficient C Y and the pitching moment coefficient C θ over the reference period obtained with the six simulations are reported respectively in the top, middle and bottom plots of Figure 14. The curves referring to the LSP calculations for the three values of M are indistinguishable, as one would expect on the basis of the aforementioned theoretical and physical evidence. The three curves of the force coefficients computed without LSP for M = 0.1 are also equal to their counterparts obtained with the LSP-enhanced simulations. However, the two sets of curves ob- 42

58 tained without LSP for M = 0.01 and M = differ from the other four sets, with such differences being larger for M = These differences emphasize the loss of accuracy incurred by not using LSP when solving complex low-speed air flows, like that being considered, by means of a compressible NS solver. In order to further investigate this accuracy problem, the instantaneous static pressure and vorticity fields observed when the wing is at 25 % of the reference period (position 1) are examined in Figure 15 and when the wing is at % of the reference period (position 2) are Figure 15: Comparison of static pressure coefficient (C p ) and normalized vorticity (Ω f ) contours for position 1 of oscillating wing for three values of M obtained with and without LSP. 43

59 Figure 16: Comparison of static pressure coefficient (C p ) and normalized vorticity (Ω f ) contours for position 2 of oscillating wing for three values of M obtained with and without LSP. examined in Figure 16. Each subplot reports the solution obtained using a given value of M with and without LSP. The top left subplot presents the comparison of the static pressure coefficient C p for M = 0.1 obtained with and without LSP, whereas the top right subplot provides the contours of Ω f, the flow vorticity normalized by the freestream velocity, obtained with the same two simulations. The mid and bottom subplots have the same structure as the top ones, but they refer to M = 0.01 and M = respectively. Examination of Figures 15 and 16 reveals that the three LSP solutions are independent of M, as neither the C p nor the Ω F contours 44