Large-Eddy Simulation and Trailing-Edge Noise Prediction of an Airfoil with Boundary-Layer Tripping

Transcription

1 15th AIAA/CEAS Aeroacoustics Conference (3th AIAA Aeroacoustics Conference) May 9, Miami, Florida AIAA Large-Eddy Simulation and Trailing-Edge Noise Prediction of an Airfoil with Boundary-Layer Tripping Julian Winkler Institut für Fluid- und Thermodynamik Universität Siegen, Paul-Bonatz-Str. 9-11, 5768 Siegen, Germany. Stéphane Moreau Groupe d acoustique de l Université de Sherbrooke (GAUS) Université Sherbrooke, 5, boul. de l Université Sherbrooke (Québec), Canada J1K R1. Thomas Carolus Institut für Fluid- und Thermodynamik Universität Siegen, Paul-Bonatz-Str. 9-11, 5768 Siegen, Germany. This paper deals with hybrid methods for trailing edge noise prediction of a single NACA airfoil at zero angle-of-attack. The procedure is based on two steps. First, an incompressible large-eddy simulaton (LES) of the airfoil trailing-edge flow is performed. Then, in a second step, the far-field acoustic pressure is predicted from the LES source terms using three different methods based on the acoustic analogy. These are Amiet s and Ffowcs Williams & Hall s trailing-edge noise theories and Curle s compact dipole solution. Aerodynamic and acoustic results are then compared to experimental measurements performed in the aeroacoustic wind tunnel of the University of Siegen. For comparison purposes with experimental measurements, the large-eddy simulation includes wind tunnel installation effects by a choice of suitable inflow boundary conditions. The experimental chord-based Reynolds number is , which results in long laminar flow regions along the airfoil that lead to Tollmien-Schlichting instabilty waves with an associated additional tonal and broadband noise radiation. To avoid this additional noise source, the boundary layer of the airfoil had been tripped on both sides in the experiment. This tripping has been included in the numerical grid of the LES in its full complex configuration (forward facing serrations) and also as a simplified geometry (simple stair-step). Eventually three different LES computations are compared that differ in boundary condition and boundarylayer tripping modeling. The modeling effects are assessed with regard to the aerodynamic and aeroacoustic prediction capability of the respective LES approaches. The comparison stresses once more the necessity of accurate boundary conditions in the LES in order to arrive at comparable results with wind-tunnel measurements. Nomenclature Roman letters C p pressure coefficient, [-] c airfoil chord length, [m] c speed of sound, [m/s] E, F combinations of Fresnel integrals, [-] E qq dimensionless wake energy spectrum, [-] erf Error function, [-] Ph.D. student. IFT, Fachbereich Maschinenbau, Universität Siegen, julian.winkler@uni-siegen.de. Professor, GAUS, Mechanical Engineering Department, Université de Sherbrooke, stephane.moreau@usherbrooke.ca. Professor. IFT, Fachbereich Maschinenbau, Universität Siegen, thomas.carolus@uni-siegen.de. 1 of 5 Copyright 9 by J. Winkler. Published by the American Institute American of Aeronautics Instituteand of Aeronautics Astronautics, Inc., andwith Astronautics permission.

2 f,f s frequency, sampling frequency, [Hz] I,I LE,I TE radiation integral in Amiet s model, [-] k = ω c acoustic wavenumber, [m 1 ] K = ω u convective wavenumber, [m 1 ] K x,k y streamwise and spanwise aerodynamic wavenumber, [m 1 ] L airfoil span, [m] L LES simulated airfoil span, [m] L x,l y,l z cell dimensions of computational grid, [-] l y spanwise correlation length of the turbulent pressure field, [m] M Mach number, [m/s] n normal vector in Curle s model, [-] p a far-field acoustic pressure, [Pa] p ref reference pressure (= 1 5 ), [Pa] Q second invariant of the velocity gradient tensor, [s ] q,q instantaneous and mean velocity magnitude, [m/s] Re c Reynolds number based on chord length, [-] R distance between source and observer in Ffowcs Williams & Hall s theory, [m] S source term in Ffowcs Williams & Hall s theory, [kg/(s m 5/ )] S convection-corrected far-field observer position, [m] S ij rate-of-strain tensor, [s 1 ] S pp far-field acoustic pressure power spectral density, [Pa /Hz] Sr Strouhal number, [-] u mean flow speed, [m/s] u c convection speed, [m/s] u r,u θ cylindrical polar velocity components, [m/s] V integration volume around the airfoil, [m 3 ] r,θ,z cylindrical polar coordinates of observer field point, [m], [rad], [m] r,θ,z cylindrical polar coordinates of source field points, [m], [rad], [m] x, y observer and source coordinates, [m] x,y,z streamwise, spanwise and crosswise cartesian coordinates on airfoil surface, [m] x 1,x,x 3 streamwise, spanwise and crosswise cartesian coordinates for observer location, [m] Symbols β compressibility parameter, [-] γ coherence function, [-] Λ integral length-scale of turbulent velocity field, [m] µ reduced frequency, [-] Ω ij rate-of-rotation tensor, [s 1 ] ω angular frequency, [rad/s] Φ wall-pressure cross spectral density, [Pa /Hz] Φ pp wall-pressure power spectral density, [Pa /Hz] Φ qq wake-velocity power spectral density, [(m/s) /Hz] φ off-midplane angle of source field points, [rad] Π(K x,k y,ω) wavenumber-frequency spectrum of wall-pressure fluctuations, [Pa /Hz] Π (K y,ω) streamwise-integrated wavenumber-frequency spectrum, [Pa m/hz] ρ fluid density, [kg/m 3 ] Θ phase of the cross-spectral density, [rad] τ time separation, [s] ξ,η streamwise and spanwise separation distance on airfoil surface, [m] ζ y constant in Corcos model, [-] Superscripts complex conjugate variables in the subcritical solution to Amiet s model non-dimensionalized by half chord-length of 5

3 ˆ Abbreviations bc FWH LE PSD rms SPL TE TS variable with temporal Fourier transformation applied non-dimensionalized by c sin φ half-plane solution boundary condition Ffowcs Williams & Hall Leading Edge Power Spectral Density root mean squared Sound Pressure Level Trailing Edge Tollmien-Schlichting I. Introduction One major noise source in many turbomachinery applications, especially wind turbines, is the production of sound by turbulent flow over the trailing edge of the blade. It is an inevitable noise source and easily dominates the broadband noise at low Mach number. Numerical prediction methods of the broadband trailing-edge noise for an airfoil at relatively high Reynolds number are feasible only, if certain simplifications are made. A common procedure for low Mach number flows is to use a so-called hybrid scheme, where the flow calculation is decoupled from the acoustic prediction: an incompressible flow calculation is made first and in a second step the sound is predicted from the source terms of the flow field by, for instance, using Lighthill s acoustic analogy. 17 An important issue for trailing-edge noise prediction is to accurately capture the turbulent boundary layer development close to the trailing edge, since it is this pressure field that is scattered into acoustical pertubations at the edge. For airfoils at relatively low Reynolds numbers ( 1 5 ), the boundary layer may still be in transition to turbulence when it reaches the trailing edge. This usually results in large laminar flow regions around the airfoil and Tollmien-Schlichting (TS) instability waves developing on the suction side or pressure side of the airfoil, which will interact with the trailing edge. These TS instability waves create an acoustic feedback loop between the point of instability onset and the trailing edge, which results in extraneous far-field noise characterized by typical broadband humps on which regular tones are superimposed. 3 In order to study the pure trailing-edge noise which is generated by the scattering of the turbulent boundary layer at the trailing edge, it is necessary to remove the additional instability noise mechanism if experiments are conducted at rather low Reynolds numbers. This can be done by artificially tripping the boundary-layer. As shown by Hama, 13 the most efficient trip is a thin serration applied close to the leading edge on both airfoil sides. In order to mimic experiments with boundary-layer tripping numerically, it is required to incorporate such tripping devices in the numerical grid. This can be done either by using the exact geometrical details of the tripping device or simplifying it in order do reduce the computational cost of complex grid topologies. The influence of geometrical details of tripping devices on the wall-pressure evolution and acoustic prediction is investigated numerically in the present study. For this purpose, a NACA airfoil has been selected. The airfoil is set to geometrical angle-of-attack (with respect to the chord length). This corresponds to an operating condition at high mass flow rate, beyond the design condition of any turbomachinery. Large-eddy simulation of the flow around the airfoil is conducted for a Reynolds number of Re c = , solving the incompressible Navier-Stokes equations. Three LES computations are performed. Two of them include different tripping models: the actual experimental boundary-layer serrations and a simplified stairstep trip of the same thickness. Both LES take wind tunnel installation effects into account by choosing appropriate inflow boundary conditions on a truncated LES-domain. Both solutions are then compared to an LES of the same airfoil in free-field conditions, but without boundary-layer tripping, for which the lift distribution disagrees with the experiment. The latter LES is the most common and simplifying approach usually found in the literature. Wall-pressure spectra and wake-velocity profiles from all three simulations are then compared with the experimental data collected in the small anechoic wind tunnel at the University of Siegen. Acoustic far-field predictions based on three different models, using Lighthill s acoustic analogy, 17 are compared with the measured sound pressure levels and modeling implications are discussed. Finally, the 3 of 5

4 influence of the three different modeling strategies are assessed from an aerodynamic and aeroacoustic point of view. II. Experimental Study Experiments have been conducted in the small aeroacoustic wind tunnel facility at the University of Siegen. The facility is equipped with a muffler and several screen and honeycomb combinations to obtain a flat velocity profile at the nozzle exit with turbulence intensities below.% (after recent improvements). It terminates in an semi-anechoic chamber to allow for acoustic measurements. Winkler & Carolus have recently provided a detailed description of the facility. 9 The NACA airfoil mock-up has a 13.5 cm constant chord length (c) and a 18 cm span (L). It is placed.56 c downstream from the nozzle exit plane and held between two horizontal side plates fixed to the nozzle of the open-jet wind tunnel. These plates are 18 cm (1.33 c) apart and the width of the rectangular jet is 18 cm. The chord-based Reynolds number of the flow is Re c = and the angle-of-attack is zero. This small incidence has been selected to limit the jet deflection and the interaction of the shear layers coming from the nozzle lips with the airfoil. This is also advantageous for the numerical simulation procedure described in Sec. III. The NACA airfoil mock-up is equipped at mid-span with 7 pressure holes which allow for lift measurements and also measurement of the wall-pressure fluctuations by flush-mounted remote microphone probes (RMP). Figure 1 (left) shows the layout of the streamwise pressure measurement locations at the midspan plane of the NACA airfoil. The airfoil wake has been measured with a 3-D hot-wire (a double x-wire sensor, sometimes referred to as Kovásznay-probe). Measurements were taken in eight planes downstream from the trailing edge, Fig. 1 (right). For each plane more than 5 points have been selected for velocity measurements. Acoustic measurements have been made by three microphones at a distance of 1. m from the trailing edge under different observer angles. A cross-correlation technique 5,9 between the microphone signals has been implemented to extract the trailing-edge noise from the facility background noise. It has been found that at the present Reynolds number, Tollmien-Schlichting instability waves occur in the laminar boundary layer which lead to additional sound radiation, including broadband and tonal noise components. In order to suppress these instabilities and the associated sound radiation, the boundary layer of the airfoil had been tripped on both sides at 1% chord by a (forward) serrated aluminum tape as suggested by Hama, 13 see Fig. 1 (left). This trip has been varied in thickness by multiple layers of each 8 µm. This procedure allows to capture in very small steps the minimum, but yet sufficient, disturbance to the boundary layer to obtain transition for a given Reynolds number and angle-of-attack...1 y/c y/c x/c Figure 1. Left: Instrumented airfoil for experimental measurements; right: Locations for wall-pressure and lift measurements on the airfoil surface (top) and hot-wire measurement planes in the wake (bottom). III. Large-Eddy Simulation Three different LES are performed in order to study the modeling issues of the boundary-layer tripping and the wind tunnel installation effects on the aeroacoustic prediction performance. Two simulations include a boundary-layer trip in the grid, taking into account the wind tunnel installation effects, whereas the third 4 of 5

5 simulation calculates the flow around an untripped airfoil at freeflight conditions. The simulation cases are denoted from now on by serration trip LES, for the LES including the exact boundary layer trip geometry of the experiment; step trip LES, for a simplified boundary layer trip geometry; and freeflight LES for the LES without any boundary-layer tripping and without incorporating the wind tunnel installation effects. Details on the flow solvers, numerical setup, boundary conditions and grid topologies are given in the next three sections. III.A. Flow Solvers The LES are based on the spatially filtered, incompressible Navier-Stokes equations with the dynamic subgrid-scale model. 1,18 They are solved with two commercial codes, Ansys R CFX V11 and Ansys R Fluent V6.3. In the former, the governing equations are solved with a control-volume-based finite element method. Central differencing is used for the spatial discretization. Time integration is made with an implicit scheme. In the latter, the governing equations are solved with a classical finite-volume method. A central-difference scheme is similarly used for the spatial discretization and the non-iterative time-advancement (NITA) scheme is used for the time advancement. Therefore in both cases, solutions are second-order accurate in space and time. Preliminary RANS computations are required for generating the LES boundary conditions for two of the three simulations, as described in the next section. They have been performed using the Shear-Stress- Transport (SST) turbulence model developed by Menter 19 and implemeted in Ansys R CFX V11, with again second order accurate solutions for all variables. III.B. Numerical Setup and Boundary Conditions As shown by Moreau et al., the flow around an airfoil when installed in a free-jet wind tunnel significantly deviates from that of the same airfoil placed in a uniform stream. The importance of accounting for these installation effects is more pronounced in the current study as the wind tunnel jet dimension is relatively small compared to the mock-up size and airfoil aspect ratio. The computational procedure pursued here is the same as recently used by Winkler & Moreau, 8 as well as Christophe & Moreau. 7 In a first step, the complete wind-tunnel setup (Fig., left) is simulated by a RANS computation, including the nozzle and the airfoil. Measured velocity profiles are used for the nozzle inlet boundary conditon. From this solution, the correct boundary conditions are extracted for a simplified run on a truncated domain (Fig., right top). Second, a RANS computation is performed on the truncated domain (grid extracted from the full domain, as indicated in Fig. ), using the correct inflow boundary conditions obtained from the first run. Third, the truncated RANS domain is extruded in the spanwise direction and a full 3-D LES is performed on the airfoil, again using the RANS inflow conditions. The computational domain size for the LES has been chosen so that the nozzle shear layers do not have to be resolved in the grid. The airfoil is therefore ideally placed in the potential core of the jet. To avoid intrusion of the nozzle shear layers also further downstream, the computational domain of the airfoil wake region has been rotated by the flow deflection angle, which is known a priori from experiments (Fig., right). The LES domain size is 3.5 c in the streamwise (x) direction, 1 c in the crosswise (y) direction and.74 c in the spanwise (z) direction. This holds for two of the three LES (serration trip and step trip LES). Both of these LES use the steady RANS velocity profiles (u and v components) as inflow conditions along the whole extended C-contour (Fig., right), a no-slip boundary condition on the airfoil surface, a convective outflow boundary condition at the wake exit and periodic or symmetric conditions in the spanwise direction. The third LES, (Fig., right bottom) follows the usual strategy: A C-Contour around the airfoil profile is used with a a constant velocity inlet boundary condition. Therefore, this simulation mimics the airfoil in freeflight conditions. The domain size is 5.5 c in streamwise, 5 c in crosswise and.1 c in spanwise direction. The whole computational wake region is again rotated by the known wake-deflection angle, but wind-tunnel effects are not taken into account. The chord-based Reynolds number for all cases is Re c = The size of the time step in all three LES has been chosen so that the convective Courant-Friedrichs-Lewy number is about 1, i.e., t = s. The simulations have run for approximately 1 flow-through times, based on the freestream velocity and airfoil chord length, before the solutions reached a statistically steady state. 5 of 5

6 Figure. Left: Grid topology of the wind tunnel setup for initial RANS computations. Right top: LES truncated domain, extracted from the full wind-tunnel setup and used for the two boundary-layer tripped LES. Right bottom: Grid for the freeflight LES, mimicking the airfoil in free-field conditions and without boundary-layer tripping. III.C. Grid-Topology For the serration trip and the step trip LES, a block-structured C-mesh is used, shown in Fig., right top, with L x L y L z = (serration trip LES) and (step trip LES) cells along the airfoil. A total of 3 streamwise grid points are placed in the wake region. The near-wall resolution is y A smooth grid distribution and orthogonality at the wall is applied and the grid-stretching is limited in both the streamwise and crossflow directions to ensure numerical stability. To model the boundary-layer trip, two cases have been set up. One including the realistic boundary-layer tripping, which is a forward-serrated aluminum tape, having dimensions as used in the experiments (thickness c). This trip essentially creates a 3-D disturbance in the flow. The region above and behind the trip has been resolved on a finer scale in the spanwise direction with 8 grid points ( = ) to better capture the expected transition process. The finer resolved blocks have a streamwise extent of 19 trip thicknesses and a height of 19 trip thicknesses. The remaining grid has a spanwise resolution of = At the interfaces of the refined grid region with hanging nodes (see Fig. 3, left), the CFL number can locally reach values on the order of 1. The grid topology of the second run with a simplified step replacing the serrations, but of the same height and length (creating a -D disturbance) is shown in Fig. 3 (right). No spanwise refinement is made in that case. The third simulation (freeflight LES) describes the airfoil in the usual manner: neither wind tunnel installation effects nor boundary-layer tripping are included. The airfoil is surrounded by a C-mesh of larger extent than in the other two simulations. The spanwise extent is slightly larger with =.1 and has 64 grid points along the span. The spanwise resolution is thus = Figure 3. Left: Grid refinement on the airfoil for resolving the boundary-layer transition process for the serration trip LES. Middle: Close-up view on the serration boundary-layer trip mesh region. Right: Grid topology of the simplified step trip. 6 of 5

7 III.D. Summary of Simulation Parameters The specifications given in the last three sections are finally summarized in Table 1. Case serration trip LES step trip LES freeflight LES x y z 3.5 c 1 c.74 c 3.5 c 1 c.74 c 5.5 c 5 c.1 c L x L y L z (airfoil) (8) L x L y L z (wake) total no. of cells subgrid-scale model dynamic Smagorinsky dynamic Smagorinsky dynamic Smagorinsky spanwise bc periodic symmetric periodic inflow bc RANS velocity profile RANS velocity profile constant velocity boundary-layer trip yes yes no solver Fluent V6.3 CFX V11 CFX V11 Table 1. Summary of cases investigated. IV. Trailing-Edge Noise Prediction Methods Trailing-edge noise computations for all simulations are performed by three different methods based on Lighthill s 17 acoustic analogy. The difference being the source terms used for the acoustic prediction. Whereas Ffowcs Williams & Hall s 11 trailing-edge noise theory requires a volume integral over the velocity field around the trailing edge, Amiet s and Curle s 9 theory utilize some information of the surface pressure on the airfoil, therefore requiring surface integrals at most. The basic equations and assumptions for all three methods are given in the next subsections. IV.A. Ffowcs Williams and Hall s Trailing-Edge Noise Model The formulation by Ffowcs Williams & Hall 11 simplifies the airfoil to a semi-infinite flat plate with zero thickness. Lighthill s equation is then solved using the exact Green s function for a scattering half-plane. For sources close to the trailing edge (within one wavelength), a series expansion is used to approximate the Green s function. Based on these ideas, the far-field acoustic pressure at a given observer location x(r,θ,z) and for a given angular frequency ω is given by p a ( x,ω) = e i π 4 k sin θ π 1 { ) ρ (û θ û r V e i k R (sin φ) 1 4π R (k r ) 3 sin θ ρ u r u θ cos θ } d 3 y (1) where the caret denotes temporal Fourier transform and V is the integration volume around the airfoil trailing edge. The velocity components u r and u θ are defined in a cylindrical polar coordinate system around the trailing edge. The vector y(r,θ,z ) represents the source-field points with R = x y and sin φ = r/[r + (z z ) ] 1. Wang & Moin 6 have provided a simpler expression, that can be computed directly on-the-fly during the LES. This expression is applicable if the spanwise extent of the source field is small relative to the observer distance, which is always fulfilled when the observer is in the acoustic far-field. The acoustic pressure is then given by the following expression: p a ( x,ω) ei(k x π/4) 5 π 3 x (k sin φ) 1 θ sin () Ŝ(ω) with S(t) = V ρ r 3 [ (u θ u r)sin θ u ru θ cos θ ] d 3 y (3) 7 of 5

8 Howe 15 has proposed a finite-chord correction to the exact half-plane Green s function. This correction is obtained by applying a multiple scattering procedure at the leading and trailing edge. The resulting far-field acoustic pressure including the finite-chord correction factor is given by 7 p a ( x,ω) = 1 + ( e i k F k π π k ) sin θ e i π 4 e i k cos θ F ( ) e i k + ei k sin θ πi k ( k π ) cos θ p a ( x,ω) (4) where F is a combination of the Fresnel integral auxiliary functions g and f: F(x) = g(x) + if(x), defined in Abramowitz & Stegun, 4 7.3, and k = k csin φ. IV.B. Amiet s Extended Trailing-Edge Noise Model Amiet s trailing edge noise theory is similar to the Ffowcs Williams & Hall 11 approach: it simplifies the airfoil to a semi-infinite flat plate. However, it treats trailing-edge noise as a diffraction problem, where an incident turbulent boundary-layer pressure gust is scattered at the edge. A finite-chord correction, which takes into account the back-scattering effect of the pressure waves from the trailing edge at the leading edge, has been derived by Roger and Moreau. This correction, valid for any Mach number, is basically equivalent to Howe s 15 correction (Eq. 4) for very low Mach numbers. The acoustic pressure in terms of its power spectral density (PSD) in the midspan plane at a given observer location x = (x 1,x,) = (r,θ,z = ) and for a given angular frequency ω is calculated by S pp ( x,ω) = p a ( x,ω)p a( x,ω) = ( ωx3 L 4πc S + ) c Π (ω, Ky )sinc [ L c ( K y k x )] ( ) ω I, Ky d S u K y (5) c where S = x 1+β (x +x 3) is the convection-corrected far-field observer position and I is the radiation integral that involves the aerodynamic response to the incident pressure gust, including the back-scattering effect from the leading edge. It is given in the appendix. Π is the streamwise-integrated incident wavenumberfrequency spectrum, which is obtained from the full wavenumber-frequency spectrum Π by Π(K x,k y,ω) = 1 (π) + + Π (ω,k y ) = + Φ(x,y,ξ,η,ω)e i(kxξ+kyη) dξ dη (6) Π(K x,k y,ω) dk x (7) with angular frequency ω, streamwise wavenumber K x and spanwise wavenumber K y. Φ is the wall-pressure cross-spectral density, defined as Φ(x,y,ξ,η,ω) = 1 π + p (x,y,t)p (x + ξ,y + η,t + τ) e iωτ dτ (8) with streamwise and spanwise space separation ξ and η between two points on the airfoil surface and time separation τ. The amount of input required in Eq. 7 is huge, but available from the LES. Experimentally this information is difficult to obtain, since it requires a large array of pressure sensors. For practical purposes, Corcos multiplication hypothesis 8 is assumed to hold a, which states that the full cross-spectral density of wall pressure fluctuations can be separated into two multiplicative functions, one describing the streamwise (longitudinal) correlation and the other the spanwise (transverse) correlation of the pressure spectrum. This approach yields the following simplification: + Π(K x,k y,ω) dk x = + + Φ pp (ω) 1 + π γ(ξ,,ω)eiξ(kx ω/uc) dξ 1 π γ(,η,ω)eikyη dη dk x a Even though it has been shown by some investigators that it is not exactly fulfilled. 5 8 of 5

9 + = Φ pp(ω) γ(,η,ω)e ikyη dη π = Φ pp(ω) l y (K y,ω) (9) π with K x = ω u c for frozen turbulence. Φ pp is the wall-pressure power-spectral density or point pressure frequency spectrum, which is derived from the wall-pressure cross-spectral density by setting ξ = and η = in Eq. 8. l y is the spanwise correlation length near the trailing edge: l y (K y,ω) = γ (η,ω) cos(k y η)dη (1) and γ is the coherence between two points on the airfoil surface separated by ξ and η: γ (x,y,ξ,η,ω) = Φ(x,y,ξ,η,ω) Φ(x,y,ω) Φ(x + ξ,y + η,ω) (11) At a fixed chordwise location x = const., ξ = and the spanwise coherence function for a homogeneous turbulent field in y and frozen in x is γ (η,ω) = Φ(η,ω) Φ pp (ω) Φ pp (ω) (1) which is the expression used in Eq. 1. Amiet s model accounts for the diffraction of an incident pressure field on one side of a flat plate. In order to obtain the scattered field from turbulent flow on both sides, Eq. 5 is applied twice, using wall-pressure spectra and integral length-scales from suction side and subsequently from pressure side, which are assumed to radiate independently from each other. IV.C. A Note on the Unsteady Kutta Condition Up to the present day, it is still a point of controversy in how far the unsteady Kutta condition is fulfilled at the trailing edge. The Kutta condition states the the pressure difference between bottom and top side at the trailing edge is finite and equal to zero. Howe 14 already pointed out in 1978 that the application of the Kutta condition significantly diminishes the sound radiation from the trailing edge. In his trailing edge noise theory he added a convection speed of the shed secondary vortices into the wake. That velocity was an open parameter to choose. Essentially, if the shed vortices convect at the same speed as the incident pressure gusts, there would be no sound radiation at all, whereas in the case of no Kutta condition, the wake velocity would be set to zero. The difference can amount up to 1 db in the predicted far-field sound. Thirty years later, this issue still remains unresolved. Recent direct numerical simulations by Sandberg et al. indicate that the Kutta condition is approximately satisfied for a flat plate 3 at zero angle-of-attack, whereas for a NACA 1, 4 it seems to depend on the angle-of-attack and airfoil thickness at the trailing edge. In the Ffowcs Williams & Hall 11 trailing edge noise model, the Kutta condition is not satisfied. The Green s function has a singularity at the trailing edge, which is evident by inspecting Eq. 1. This singularity, however, has not caused any problems in the application of the model to the present simulations, since the velocity in close proximity to the trailing edge is very low, so that a singularity in the source term S(t) is actually never attained. On the other hand, Amiet s trailing edge noise model, Eq. 5, satisfies the Kutta condition by imposing the boundary condition that the incident and scattered pressure must cancel in the wake. Even in this case the precise formulation of the Kutta condition is ambiguous. Roger & Moreau impose the Kutta condition on the incident pressure alone, whereas Sandberg et al., 4 as well as Zhou & Joseph 3 apply the Kutta condition to the pressure jump. In the latter approach the predicted far-field sound is 6 db less than that of Roger and Moreau, which in turn is in the asymptotic limit in agreement with Howe s 14 (for wake shedding velocity of zero) theory as well as numerous experiments conducted at Ecole Central de Lyon. 1 In the present study, the Kutta condition is imposed on the pressure jump across the airfoil for reasons to follow (Sec. VI). This is also equivalent to the statement of Howe 14 that the incident pressure (Φ pp (ω) in Eq. 9) reduces to half of its value at the trailing edge, since its specular reflection vanishes, which in turn is accomplished by the pressure scattering process at the edge. 6 9 of 5

10 IV.D. Curle s Solution for a Compact Chord Airfoil The analogy by Curle 9 uses the free-space Green s function to give an integral solution to Lighthill s equation. In the presence of a rigid airfoil at low Mach number, the far-field acoustic pressure p a only involves a surface integral of the wall-pressure fluctuations. Furthermore, if the airfoil is assumed to be compact and the observer in the far-field, retarded time variations can be neglected and p a reads p a ( x,t) 1 R i 4πc S R ( nj p ij) t (t Rc ) d y (13) where R is the distance between the source on the airfoil ( y) and the observation point ( x). The contribution from the viscous stress tensor has been neglected. One limitation to Curle s approach is that it is strictly valid only for acoustically compact bodies, i.e., for acoustic wavelengths much larger than the airfoil chord, if the simulation is incompressible. If the body is non-compact, its surface extends into the acoustic far-field and the surface pressure can no longer be treated as incompressible. In that case, a tailored Green s function is required to obtain the correct solution. Therefore it is expected that Curle s solution will yield realistic prediction results only in the low to medium frequency range. V.A. V. Aerodynamic Results Mean and Fluctuating Pressure Distribution The mean pressure distributions from all three LES and a preliminary RANS computation are compared with the experimental ones in Fig. 4 (left). The RANS data (only the result from the whole wind tunnel setup is shown, the truncated 3-D domain gives almost identical values and is therefore omitted) are seen to agree fairly well with the measured pressure distribution. However, the separation occuring at around x/c =.8 on the suction side is not captured. The LES with serrations on the other hand captures this separation, but slightly deviates in the overall level of the pressure coefficient on the suction side for x/c <.7. The simplified step trip LES predicts the separation point to be further downstream and the C p agree much better with the experiment on the suction side. However, additional separation is observed on the pressure side, which is neither evident from the experiment nor from the serration trip LES. This could be attributed to the insufficient tripping effect of the step located just upstream of this separation bubble. The local experimental pressure peaks appearing in Fig. 4 (left) are caused by the boundary-layer trips on both airfoil sides. The freeflight LES gives a C p distribution that deviates significantly from the experimental one because of the restricted wind tunnel jet and mean flow deflection that is not taken into account in the freeflight LES. Therefore, the lift corresponds to a higher angle-of-attack as compared to the one in the experiments and the other two LES. Nevertheless, the chordwise location of the local separation on the suction side is in close agreement with the serration trip LES and the experiment, which was carried out with just this trip. However, an additional separation, similar to the step trip LES, occurs on the pressure side at.5 c. In summary, the serration trip LES provides the best overall agreement to the experiment. The trace of the chordwise pressure fluctuations on the airfoil, shown in Fig. 4 (right), reveals the effect of the boundary-layer tripping. For the serration trip LES, the root mean squared (rms) values of the pressure increase drastically on the pressure side in the region of the boundary-layer trip (.1 c), leading to a turbulent boundary layer for x/c >.5. On the suction side however, the tripping does not induce direct transition. The rms-values increase sharply at x =.7 c, followed by a mild separation and reattachment of the boundary layer. The peak occuring at.8 c agrees with the local loss of lift seen in Fig. 4 (left). For x >.9 c the boundary layer is turbulent and attached. In the simplified step case, the overall rms-levels are predicted to be much higher in the regions of flow separation. While the freeflight LES in general is in closer agreement with the serration trip LES in terms of the magnitude of the pressure fluctuations, the flow separation regions yield slightly higher levels in general than in the serration trip LES. The boundary-layer transition on the pressure side can be explained in light of the stronger pressure gradients near the leading edge appearing on this side, compared to the suction side. This is a result of the wind tunnel installation effects, inducing a slightly negative incidence for geometrical angle-of-attack. To further investigate the difference in the boundary-layer behavior in all three LES, flow visualization using a vortex identification technique can be used. The Q-criterion, first introduced by Hunt et al. (see Alfonsi 1 ) seems to be a suitable choice for visualizing coherent structures in the present case of a wall-bounded 1 of 5

11 C p Experiment.6 RANS wind tunnel serration trip LES.8 step trip LES freeflight LES x/c p rms /p dyn serration trip LES step trip LES freeflight LES pressure side suction side x/c Figure 4. Left: Mean wall-pressure coefficient C p on the airfoil. Right: LES predicted root mean squared (rms) values of the pressure fluctuations along the chord for the serration trip LES (top), the step trip LES (middle) and the freeflight LES (bottom). flow. Q is the second invariant of the velocity-gradient tensor and is defined as Q = 1 (Ω ijω ij S ij S ij ) (14) with the rate-of-rotation-tensor Ω ij and the rate-of-strain-tensor S ij, defined as Ω ij = 1 ( ui + u ) i ; S ij = 1 ( ui u ) i x j x j x j x j High positive values of Q identify regions of high vorticity and low shear-strain rates and regions of high negative Q indicate low vorticity and high shear-strain rates. Therefore, rotational motions can be distinguished from non-rotational motions. Figure 5 shows snapshots of the flow-field for the two LES with boundary-layer tripping for isosurfaces of Q, i.e., Q = s in a statistically stationary state of the flow solution. The following conclusions can be drawn from these figures: Both, the serration trip LES and the step trip LES do not induce a direct boundary-layer transition on the suction side. In the serration trip LES, the boundary layer develops -D Tollmien-Schlichting (TS) instability waves that occur as rolled up two-dimensional vortical structures, after an initial stable state. These disturbances are superimposed by secondary 3-D instabilities and lead to a turbulent boundary layer close to the trailing edge. The step LES stays laminar over a mucher longer region and then starts to separate. The separated boundary layer becomes unstable and turns turbulent as it reattaches close to the trailing edge. The boundary-layer transition on the pressure side of the airfoil is visualized in Fig. 5 (right), by plotting only over half of the simulated span to better illustrate the position and shape of the trip. From this it can be seen that the serration trip LES provokes an early transition by the appearance of induced oblique vortex patterns in front of the serrations. Despite the symmetry of the two included serrations in the simulation, a preferred direction for these oblique structures is evident (which is the same for the other half of the span that is not visualized in Fig. 5 (right)). The boundary-layer remains fully attached behind the serrations. For the step trip LES, Fig. 5 (right) shows that a little further downstream of the step trip, the boundary layer separates and then quickly turns into a turbulent state. In comparison, the serrations clearly add a 3-D disturbance to the flow and are found to be more effective for inducing transition. However, on the airfoil suction side, neither of the two tripping models disturbs the boundary layer sufficiently. Only at the very aft portion of the airfoil (.9 c) the boundary layer becomes turbulent. This flow visualization confirms the observations made in Fig. 4. A sufficient boundary-layer tripping on the suction side can be achieved by increasing the trip thickness. However, experiments have shown that the present trip is sufficient to avoid significant TS-waves causing tonal (15) 11 of 5

12 noise with additional broadband components. It was furthermore checked experimentally by stethoscopic measurements that the transition point is truly far downstream on the suction side and immediate on the pressure side. Yet this minimum trip thickness makes the accurate simulation more challenging. Figure 5. Visualization of the airfoil suction side (left), its trailing-edge region (middle) and the pressure side (right) of the flow-field by isosurfaces of Q = s for the serration trip LES (top) and the step trip LES (bottom). Isosurfaces are color-scaled by the magnitude of vorticity. The precise position of the flow separation and transition is shown in Fig. 6 as an instantaneous snapshot of the wall-shear stress. Separated regions are those with a wall-shear stress of zero. In the color-scaling they appear as white regions on the airfoil. The figure shows that on the suction side the flow separates at about the same location in the serration trip and freeflight LES, but separates and reattaches much further downstream in the step trip LES. On the pressure side, the serration trip LES remains fully attached and directly turns turbulent in the region near the trip. The step trip on the other hand gives rise to a separation bubble on the trip that extents to x/c =., whereas the freeflight LES yields a delayed flow separation for x/c =..4. A time-averaged plot of the velocity field shows that the serration trip and freeflight LES exhibit an attached flow, whereas the step trip LES is separated in the two indicated regions, wets the surface for x/c >.8 on the suction side and has a rather organized attached flow for x/c >.9. This separation behavior may at least partly explain the higher amplitudes observed in the pressure trace of Fig 4. Figure 6. Instantaneous contours of wall-shear stress on the airfoil suction side (left) and pressure side (right), for the serration trip, step trip and freeflight LES (from top to bottom). Scaling ranges from Pa (white) to 4 Pa (black). The space-time correlation, i.e., the inverse Fourier transform of the wall-pressure cross spectral density, Eq. 8, at selected streamwise locations is shown in Fig The serration trip LES, Fig. 7, shows the structure of the pressure field in the transitional zone of the boundary layer, as well as close to the trailing edge, for both pressure side (left) and suction side (right). The pressure side shows a periodic pattern of 1 of 5

13 single frequency near the serrations that appears as isolated correlation regions in-between uncorrelated areas. Right in front of the trip x/c =.5 the boundary layer feels the disturbance imposed by the downstream serrations. This gives rise to a periodic pattern at an oblique angle, as already shown in the iso-q contours of Fig. 5. At the end of the serrations x/c =.1 the pattern has been redirected to a streamwise periodic disturbance of certain frequency. Beyond this point the boundary layer is fully turbulent, with increasing correlation furher downstream, due to the mild adverse pressure gradient. This is concluded by comparing the chordwise locations x/c =.75 and x/c =.95 in Fig. 7, left. On the airfoil suction side (Fig. 7, right), the boundary layer transition starts approximately at x/c =.6, with a spanwise highly coherent pattern at x/c =.75, that is still present at x/c =.85, but of less spanwise coherence, until it vanishes completely at x/c =.95. The boundary layer is fully turbulent at that point and exhibits smaller structures than on the pressure side at the same location..14 x/c =.5 x/c =.1 x/c =.75 x/c = x/c =.6 x/c =.75 x/c =.85 x/c =.95 tu /c tu /c Figure 7. Serration trip LES: Average isocontours of wall-pressure space-time correlations at selected x/c locations. Left: Pressure side; right: Suction side. Contour values are from.1 to.9 with increment.1. Averaging performed timewise and 16 spanwise. Results for the step trip LES (Fig. 8) are presented for different chordwise locations, due to the different appearance of the transitional zones. On the airfoil pressure side (Fig. 8, left), boundary-layer transition is delayed to a location of x/c. with a less strong spanwise coherence, but larger structures. The boundary layer is turbulent for x/c >. and again, due to the adverse pressure gradient the turbulent structures increase towards the trailing edge. On the airfoil suction side, big structures visible at x/c =.8, turn into highly coherent quasi -D periodic structures that are the strongest in the region x/c =.85, diminish until x/c =.93 and then disappear at x/c =.95. In comparison to the serration trip case, the structures are much bigger on both sides. The freeflight LES, Fig. 9, shows a periodic flow pattern on the pressure side at x/c = as well, but is less coherent in spanwise direction and the periodicity is rather weak, as seen in the drop of the isocontour levels close to the main peak. This is the region where the flow separates from the airfoil and reattaches. Beyond x/c =.5 the flow is turbulent and fully attached. The size of the compact structures does not change significantly from x/c =.6 to x/c =.95. On the suction side, the transition region is around x/c =.7...8, accompanied by a periodic flow pattern. For x/c >.8 the flow structures are compact, indicating a fully turbulent flow near the trailing edge. Selected point pressure frequency spectra are shown in Fig. 1 and Fig. 11. They have been obtained by discrete Fourier transformation using a Hanning window and have been averaged eight times to yield a frequency resolution of f = Hz, Hz and Hz (serration trip, step trip and freeflight LES). Figure 1 (left) provides the spectral content of the pressure side signals. Narrow-band humps and their harmonics are seen when the observer point is in the transitional part of the boundary layer (which is at different locations in all three LES). They are especially pronounced in the case of the serration trip LES on the pressure side, directly behind the serrations x/c =.15 and correspond to the oblique pattern visible in Fig. 5 (top-right). These single frequencies presumably coming from boundary-layer instabilities have been characterized experimentally by Arbey & Bataille 3 among others, as well as recently with LES by Kim et al. 16 and with DNS by Desquesnes et al. 1 On the airfoil suction side these discrete components exist 13 of 5

14 .14 x/c =. x/c =.3 x/c =.6 x/c = x/c =.8 x/c =.85 x/c =.93 x/c =.95 tu /c tu /c Figure 8. Step trip LES: Average isocontours of wall-pressure space-time correlations at selected x/c locations. Left: Pressure side; right: Suction side. Contour values are from.1 to.9 with increment.1. Averaging performed 4 timewise and 17 spanwise..61 x/c =.45 x/c =.5.14 x/c =.6 x/c = x/c =.7 x/c =.8.14 x/c =.85 x/c =.95 tu /c tu /c Figure 9. Freeflight LES: Average isocontours of wall-pressure space-time correlations at selected x/c locations. Left: Pressure side; right: Suction side. Contour values are from.1 to.9 with increment.1. Averaging performed 1 timewise and 3 spanwise. approximately in the region.6 < x/c <.86 for the serration trip LES;.6 < x/c <.8 for the freeflight LES and in a more restricted domain of.83 < x/c <.93 for the step trip LES. Beyond this range the boundary layer is turbulent and free of tonal components, as already indicated in Fig Figure 11 finally shows the state of the wall-pressue spectra at x/c =.95 for the pressure side (left) and the suction side (right). The overall agreement in shape and amplitude between serration trip LES and freeflight LES is found to be surprisingly good for the pressure side and in reasonable agreement for the suction side. Approaching the trailing edge further (not shown here), say at x/c =.98, the suction side spectra almost collapse, again showing surprisingly good agreement, despite the differences in the modeling approach of both LES. The step trip LES on the other hand deviates from the other two LES at low to medium frequencies, leading to much higher levels in excess of 7 db to 15 db. At high frequencies (> 1 khz on the pressure side and > khz on the suction side), the spectral levels of all three LES are in very good agreement. The direct comparison between pressure side and suction side wall-pressure spectra reveals that suction side pressure fluctuations are much larger than their respective counterparts on the pressure side, by 7 db for the serration trip and freeflight LES and 16 db for the step trip LES. At locations close to the trailing edge, the coherence function γ in the spanwise direction is an important parameter for trailing-edge noise prediction using Amiet s method (see Sec. IV.B). It also reveals if the computational domain size is large enough, which is important for acoustic predictions of long span bodies in general (see Sec. VI). Figure 1 shows the spanwise distribution of the coherence function, Eq. 1, for 14 of 5

15 1 (pressure side) 1 (suction side) 1log(Φ /p ) [db/hz] pp ref serration trip LES at x/c =.15 step trip LES at x/c =. freeflight LES at x/c = log(Φ /p ) [db/hz] pp ref serration trip LES at x/c =.7 step trip LES at x/c =.85 freeflight LES at x/c = Figure 1. Wall-pressure spectra with reference to 1 5 Pa, at various x/c locations close to transition or separation of the boundary layer. 1log(Φ /p ) [db/hz] pp ref serration trip LES step trip LES freeflight LES x/c =.95 (pressure side) log(Φ /p ) [db/hz] pp ref serration trip LES step trip LES freeflight LES x/c =.95 (suction side) Figure 11. Wall-pressure spectra with reference to 1 5 Pa, at x/c =.95 for the airfoil pressure side (left) and suction side (right). all three LES for frequencies up to 1 khz. It is found that at lower frequencies (< 8 Hz) the spanwise domain of the serration trip (left) and step trip (middle) LES is not suffciently large to allow for a full decay of the spanwise coherence. This means that the low-frequency components of the boundary layer are correlated over a larger region than is apparently the case at higher frequencies. In the mid- and high-frequency range the coherence decays relatively fast in spanwise direction. The freeflight LES not only resolves the spanwise domain better (with 64 cells, instead of 3), but also captures a larger region (.1 c instead of.74 c) than the other two LES. The coherence is found to drop to zero within the simulated domain. The airfoil section therefore comprises a statistically independent acoustic source region that can be understood as radiating sound incoherently from other spanwise distributed sources of the real airfoil span. The implications of this will be addressed in Sec. VI. The convection speed u c of the turbulent boundary layer is required as an input parameter into Amiet s trailing edge noise model, Eq. 5. It can be obtained from the phase between the wall-pressure cross-spectrum of two streamwise locations in proximity to the trailing edge. The phase for two points separated by ξ in streamwise direction at a fixed spanwise location z, is defined as: ( ) I(Φ(x + ξ,z,ω)) Θ(x,x + ξ,ω) = arctan (16) R(Φ(x + ξ,z,ω)) 15 of 5