Direct numerical simulations of low Reynolds number flow over airfoils with trailing-edge serrations

Journal of Sound and Vibration 330 (2011) 3818–3831

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Direct numerical simulations of low Reynolds number flow over airfoils with trailing-edge serrations$ R.D. Sandberg , L.E. Jones Aerodynamics and Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK

a r t i c l e in f o

abstract

Article history: Accepted 6 February 2011 The peer review of this article was organised by the Guest Editor Available online 25 February 2011

Direct numerical simulations (DNS) have been conducted of NACA-0012 with serrated and straight flat-plate trailing-edge extensions using a purposely developed immersed boundary method. For the low Reynolds number airfoil flows accessible by DNS, laminar separation bubbles involving laminar-turbulent transition and turbulent reattachment occurs. Comparing results from simulations with serrated and un-serrated trailing-edge extensions, noise reduction for higher frequencies is shown using power spectra and one-third octave averaged pressure contours. The effect of the trailing-edge serrations on an acoustic feedback loop observed in previous simulations and the subsequent effect on the laminar separation bubble is studied via cross-correlations, probability density functions of skin friction and spanwise wavenumber spectra. The results show that the presence of serrations leads to some spanwise variation of transitional structures in the separated shear layer, but does not significantly affect the overall hydrodynamic field on the airfoil upstream of the serrations. Two reasons for why the hydrodynamic field is not considerably affected by the presence of serrations are suggested. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Airfoil self-noise, i.e. noise produced by the interaction of an airfoil’s boundary layers and wake with itself, is an important noise source in many applications, ranging from wind turbines and helicopter rotors to fan blades and airframes. Brooks et al. [1] classified five mechanisms for airfoil self-noise, relating all but one to the interaction of disturbances with the airfoil trailing edge (TE). This is partly due to the fact that fluctuations which encounter a sharp edge of a solid body (such as a TE) are scattered, which leads to a considerable increase in the radiated sound power (M5-scaling [2]) over fluctuations in free space (M8-scaling [3]). For that reason, TE noise is typically one of the main noise sources, particularly at low Mach numbers, and hence the development of trailing-edge noise theories has been given much attention. An important example is Amiet’s classical trailing-edge noise theory [4] which predicts the far-field noise produced by the turbulent flow over an airfoil using only the airfoil surface pressure difference as input. Although the presence of additional noise sources has been reported for certain flow conditions or airfoil geometries [5], for most practical applications the TE noise mechanism will dominate. Thus, any successful airfoil self-noise reduction measures will have to address the trailing-edge noise mechanism. A substantial amount of research, of analytical, experimental, and, more recently, numerical nature has addressed the trailing-edge noise mechanism, starting with the work by Ffowcs Williams and Hall [2]. In Amiet’s TE noise theory, the radiated sound is directly proportional to the spanwise correlation length of the turbulence, which is typically assumed to be given by Corco’s model. A change of the

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This article is an expanded, peer-reviewed version of an article previously published in Procedia Engineering, 6C, 2010, pp. 274–282.

 Corresponding author. Tel.: + 44 23 80597386.

E-mail address: [email protected] (R.D. Sandberg). 0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.02.005

R.D. Sandberg, L.E. Jones / Journal of Sound and Vibration 330 (2011) 3818–3831

g

Nomenclature A cf E f h M n p Pr Q qk R Re St t T ui W x,y,z

amplitude of instability wave skin friction total energy E ¼ T=½gðg1ÞM 2 þ0:5ui ui nondimensional frequency streamwise length of serration Mach number normal direction pressure Prandtl number second invariant of velocity strain-rate tensor heat-flux vector radius of computational domain Reynolds number Strouhal number time temperature velocity vector wake length of computational domain coordinates for streamwise, wall-normal directions and spanwise direction

Greek letters

a b

D Dl Dn d

Z y l

m x

r tik oi f

3819

ratio of specific heats distance between two grid points distance between two image points normal distance from ghost point to boundary displacement thickness lateral direction in curvilinear coordinates angle between line normal to immersed boundary and x-axis spanwise length of serration molecular viscosity tangential direction in curvilinear coordinates density stress tensor vorticity vector angle between line normal to immersed boundary and z-axis

Subscripts 1,2,3 d i,j,k min w

indices for streamwise, wall-normal directions and spanwise directions diameter of cylinder indices for Cartesian tensor notation minimum amplitude wall

angle-of-attack spanwise wavenumber

spanwise correlation length by any means of TE modifications would therefore have a direct effect on the radiated noise. The importance of correlation and coherence of fluctuating surface pressure is, for example, discussed in Wang et al. [6]. Previous experimental and analytical studies have shown that TE modifications can reduce airfoil self-noise without compromising aerodynamic performance. The addition of brushes to airfoil trailing edges, for example, was found to reduce the intensity of trailing-edge noise [7]. The noise reduction in that case is likely due to increased compliancy of the brushes weakening the diffraction effect at the airfoil trailing edge and alleviating the surface pressure difference. Alternatively, TE serrations have been considered. Howe [8] performed a numerical analysis of a flat plate with TE serrations possessing sawtooth-like profiles and predicted that the intensity of TE noise radiation could be reduced by such modifications, with the magnitude of the reduction depending on the length and spanwise spacing of the teeth, and the frequency of the radiation. It was determined that longer, narrower teeth should yield a greater intensity reduction. Oerlemans et al. [9] investigated experimentally the effect of adding such TE serrations to full size wind-turbine blades and found overall self-noise reductions of 2–3 dB without adversely affecting aerodynamic performance. Nevertheless, the precise mechanism by which this noise reduction occurs is not yet fully understood. Understanding those mechanisms could lead to improvements in serration design, and possibly the development of alternative techniques based on similar physical principles. This study mainly aims at numerically investigating the flow around airfoils with trailing-edge modifications to identify whether the hydrodynamic field, in particular the laminar-turbulent transition at low Reynolds numbers, is affected by the presence of trailing-edge serrations. Direct numerical simulation (DNS) is the preferred tool for such fundamental studies due to the absence of modelling. Compressible DNS allows an accurate representation of hydrodynamic phenomena, such as turbulence and transition to turbulence, and of the propagation of acoustic waves. Conducting direct noise simulations avoids interfacing between solution methods as required for hybrid approaches, and allows for the presence of acoustic feed-back loops [10], a capability essential for the current study. The complex geometries associated with trailing-edge modifications represent a considerable numerical challenge using high-order accuracy spatial schemes, however. For this reason a purposely developed immersed boundary representing the trailing-edge modification is employed. 2. Governing equations The DNS code directly solves the unsteady, compressible Navier–Stokes equations, written in nondimensional form as qr q þ ðruk Þ ¼ 0, qxk qt

(1)

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q q ðrui Þ þ ½rui uk þ pdik tik  ¼ 0, qt qxk

(2)

    q q p ðrEÞ þ ruk E þ þ qk ui tik ¼ 0, qt qxk r

(3)

where the total energy is defined as E ¼ T=½gðg1ÞM 2  þ 0:5ui ui . The stress tensor and the heat-flux vector are computed as   m qui quk 2 quj m qT tik ¼ þ  dik , qk ¼ , (4) Re qxk qxi 3 qxj ðg1ÞM 2 Pr Re qxk respectively, where the Prandtl number is assumed to be constant at Pr =0.72, and g ¼ 1:4. The molecular viscosity m is _ computed using Sutherland’s law [11], setting the ratio of the Sutherland constant over freestream temperature to 0:3686. To close the system of equations, the pressure is obtained from the nondimensional equation of state p ¼ ðrTÞ=ðgM 2 Þ. The primitive variables r, ui, and T have been nondimensionalized by the freestream conditions and the airfoil chord is used as the reference length scale. Dimensionless parameters Re, Pr and M are defined using free-stream (reference) flow properties. 3. Numerical setup The finite-difference code used for the current investigation is based on a code extensively used for compressible turbulence research, such as compressible turbulent plane channel flow [12], or turbulent flow over a flat-plate trailing edge [13]. The underlying numerical algorithm consists of a five-point fourth-order accurate central difference scheme combined with a fourth-order accurate Carpenter boundary scheme [14] for the spatial discretization, and an explicit fourth-order accurate Runge–Kutta scheme for time-stepping. No artificial viscosity or filtering is used. Instead, stability is enhanced by appropriate treatment of the viscous terms in combination with entropy splitting of the inviscid flux terms [12]. More recently the code was extended so that it could be applied to a C-type grid with wake connection. At the freestream boundary, where the only disturbances likely to reach the boundary will be in the form of acoustic waves, an integral characteristic boundary condition is applied [15], in addition to a sponge layer comprising a dissipation term added to the governing equations. At the downstream exit boundary, which is subject to the passage of nonlinear amplitude fluid structures, a zonal characteristic boundary condition [16] is applied for increased effectiveness. At the airfoil surface an adiabatic, no slip condition is applied. This variant of the code has been recently used for direct numerical simulations of transitional flows on full airfoil configurations [10,17]. The topology of the computational domain is illustrated in Fig. 1(a). The C-type domain of the cases presented in the current study had the dimensions of W= 5 chord-lengths from the trailing edge to the outflow boundary, R= 7.3 chordlengths from the airfoil surface to the freestream surface, and a spanwise width of 0.2 chords. The influence of domain size and grid resolution have been investigated thoroughly for the flow around a NACA-0012 airfoil at Re=5  104 and a ¼ 53 in Jones et al. [17], and the conclusions reached were used as guidelines for the current studies. In the tangential direction, 2570 grid points were used, with 1126 points clustered over the airfoil, including the trailing-edge extensions. In the lateral direction, 692 grid points were used and the spanwise domain was discretized with 96 points. For the investigation of the effect of trailing-edge serrations on the flow and acoustic fields, NACA-0012 airfoils with serrated and straight flat-plate extensions at the trailing edge were selected, as shown in Figs. 1(b) and (c). The advantage of a thin flat-plate geometry is that bluntness effects are minimized. The flow conditions were chosen as Re =5  104, M =0.4, and the incidence was set to a ¼ 53 because previous simulations under these conditions exhibited a slightly thinner turbulent boundary layer, and greater tendency toward tonal behaviour at low frequencies [10,17]. A thinner boundary layer means that the trailing-edge serrations do not have to be unduly large and the presence of tonal noise at low frequency allows for investigation of the effect of serrations on tonal noise components. The Mach number M= 0.4 was chosen to allow for time steps sufficiently large to obtain several periods of the lowest frequencies occurring in the sound field with the computing resources available. 4. Immersed boundary method For the serrated and non-serrated flat-plate trailing-edge extensions a new variant of an immersed boundary method (IBM) was developed. Immersed boundary methods are used to enforce boundary conditions at locations that do not coincide with computational grid points. Typically some reduction in accuracy, either in geometric representation of the boundary or of the discretization, occurs. However, advantages of such methods are that complex geometries may be simulated without requiring complex grid generation or treatment of metric discontinuities. Many approaches have been developed for enforcing immersed boundary conditions that are compatible with finite difference schemes that may broadly be divided into three groups [18]. In continuous forcing approaches (e.g. Goldstein et al. [19]), forcing terms that apply the immersed boundary condition are included within the continuous equations before discretization. This contrasts with discrete forcing approaches where the forcing is introduced after discretization. Discrete forcing approaches are

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η

3821

R ξ

W

R

Fig. 1. Topology of the computational domain (a); airfoil geometry with straight flat-plate TE extension (b) and serrated TE extension (c).

further subdivided into methods where boundary conditions are imposed indirectly (e.g. that of Mohd-Yosuf [20]), and those where boundary conditions are imposed directly (e.g. Ghias et al. [21]). The immersed boundary condition purposely developed for the current work is applied by directly modifying the computational stencil used for discretizing the governing equations in the vicinity of the immersed boundary. Thereby, a ‘sharp’ boundary interface can be realized, in contrast to methods that use distributed forcing functions. The accuracy of the method may be altered by selection of the derivative scheme employed at the immersed boundary itself, and in the vicinity of the immersed boundary when solving the governing equations. 4.1. Numerical details During pre-processing grid points are tagged depending on their proximity to the immersed boundary. Points that lie inside, but not adjacent to, the immersed boundary are denoted body points. Points that lie outside, but not adjacent to, the immersed boundary are denoted free-stream points. Points which lie adjacent to the immersed boundary and lie inside the closed body are denoted ghost points, and points which lie adjacent to the immersed boundary and lie outside the closed body are denoted body-adjacent points. These definitions are illustrated in Fig. 2(a). For each ghost point, the nearest point on the immersed boundary is located. A number of ‘virtual’ grid points, denoted image-points, are then constructed, normal to the immersed boundary and with equidistant spacing (Fig. 2b). These image points are used to evaluate the wall-normal derivatives of fluid variables, which will ultimately be used to enforce the immersed boundary condition as the simulation is progressed. When running a simulation with an immersed boundary, the boundary conditions ui ¼ 0,

(5)

T ¼ Tw

(6)

are applied on the immersed boundary at every Runge–Kutta substep. This is performed in the following manner. For each ghost point the derivatives of ui and T are computed in the direction normal to the immersed boundary, using the image points associated with that ghost point. Having then computed, for example, dui =dn and knowing that ui =0 on the immersed boundary, the value of ui at the ghost cell is computed as ui ¼ dui =dnDn,

(7)

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Δl

Δn

Fig. 2. Image (a) illustrates classification of grid-point type for use with the immersed boundary method, showing an arbitrary immersed boundary (—) and the associated body points (), ghost-points (), body-adjacent points (), and free-stream points (3). Image (b) shows an illustration of image points (+ ) associated with a ghost point, as well as the normal distance from the ghost point to the immersed boundary (Dn), and the image point spacing (Dl).

Fig. 3. Illustration of the six-point stencil employed when evaluating derivatives using the Carpenter scheme at body-adjacent grid points (left-hand stencil) and ghost points (right-hand stencil).

where Dn is the normal distance from the ghost point to the immersed boundary (Fig. 2a). In order to obtain values for the conservative variables at the image points illustrated in Fig. 2(b), linear interpolation of the conservative variables is employed. This necessitates that the spacing of the virtual image points, Dl must be approximately the same as the local spacing of the grid. In order to achieve this the following definition of Dl is employed:

Dl ¼ DxcosðyÞ2 þ DycosðyÞ2 þ DzsinðfÞ2 ,

(8)

where y is the angle between the line normal to the immersed boundary and the local x-axis, and f is the angle between the line normal to the immersed boundary and the local z-axis. Where derivatives are computed for body-adjacent points, derivative schemes must be modified such that unphysical data within the body region is not included. Essentially, when computing derivatives for body-adjacent cells the derivative stencil should include the adjacent ghost-point, but must not include any body points. In the current study the Carpenter [14] scheme with six-point stencils is employed in the vicinity of the immersed boundary, to maintain consistency with the body fitted boundary, hence derivatives for body-adjacent points are evaluated as the second point in the Carpenter stencil. The situation is illustrated in Fig. 3. An alternative approach would be to use a second-order central

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difference stencil for body-adjacent points, which would again ensure that the derivative stencil does not encompass any body points. Since a compressible code is employed in the current study, density must also be computed at all points, including ghost points. This is performed by evaluating the continuity equation at all ghost-points, as would occur normally. However, stencils used to evaluate derivatives at ghost-points are modified such that no body points are included, in a similar fashion to the body-adjacent points. Typically this means that ghost-points will employ one-sided stencils extending into the free-stream (Fig. 3), i.e. in the current case the ghost-points are computed using the Carpenter scheme, being treated as the first point in the scheme.

4.2. Validation of the immersed boundary method In order to test the immersed boundary method for geometries with boundaries not aligned with grid points, a simulation of the two-dimensional flow around a cylinder was conducted. A rectangular computational domain is specified, of dimensions (25,10), employing an orthogonal Cartesian grid. In the vicinity of the cylinder the grid spacing was uniform at Dx ¼ Dy ¼ 1  102 , and the grid was stretched to decrease in resolution with increasing distance from the cylinder. The flow around a cylinder at Red = 1  102 and 2  102 was computed, progressing each simulation until a periodic state was attained. Fig. 4 demonstrates that the flow was well resolved for both cases. The resultant Strouhal numbers of the vortex shedding, defined as St ¼ fd=U, were 0.17 and 0.2, respectively (Fig. 5), comparing well with the experimental data of Williamson [22] and numerical data of Zhang et al. [23] computed using a body-fitted grid.

Fig. 4. Iso-contours of vorticity for the two-dimensional flow over a cylinder represented by an immersed boundary, at Red = 1  102 and 2  102, using levels distributed over the range o ¼ 7 5.

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0.8 0.6 0.4

v

0.2 0 -0.2 -0.4 -0.6 -0.8

0

5

10

15 t

20

25

30

Fig. 5. Time-dependent v-velocity recorded at (x,y) = (1,0) for the two-dimensional flow over a cylinder represented by an immersed boundary, at Red = 1  102 (- -) and Red =2  102 (—).

1

ln (A/Amin)

0.8

0.6

0.4

0.2

0

600

700

800 Reδ∗

900

1000

1100

Fig. 6. TS-wave amplitude versus Red , plotted for the compressible reference case [24] (3), incompressible reference case [25] (B), body fitted DNS code (- - -) and DNS code with immersed boundary (—).

Bluff body flows illustrate the ability of immersed boundary methods to compute geometries not aligned with grid points. However, the computation of viscous instability growth is a more rigorous validation case for near-wall behaviour [24]. In order to assess the accuracy of the immersed boundary method presented here, viscous instabilitywave growth within a Blasius boundary layer was considered. A rectangular domain was specified, of dimension (4.24; 0.15), employing a Cartesian grid that was uniform in x, and non-uniform in the y-direction. The grid resolution was Dx ¼ 1:5  102 , and Dy ¼ 2:5  104 at y= 0, stretching to Dy ¼ 3  103 . A Blasius boundary layer profile was specified at  the inflow, with d ¼ 3:53  102 , Reynolds number Red ¼ 1:40  102 and M =0.25. The boundary layer was subject to periodic volume forcing via the addition of source terms to the momentum equations, at (x,y)= (0.4,0.01), introducing hydrodynamic disturbances which initially decay before amplifying within the boundary layer. In order to test the immersed boundary method, the flat-plate surface was specified at y=1.125  10  3, lying between grid points five and six in the wall-normal direction, located at y= 1  10  3 and 1.25  10  3, respectively. The maximum u-velocity amplitude across the boundary layer was determined as a function of x, and plotted as logðA=Amin Þ, where Amin is the minimum amplitude u-velocity observed across the flat plate. Fig. 6 illustrates logðA=Amin Þ for the test case plotted alongside the equivalent body-fitted results using the same code, as well as the reference incompressible DNS data of Fasel and Konzelmann [25], and reference compressible data [24]. It can be seen that the body-fitted DNS code exhibits slightly smaller growth rates than the incompressible reference data, but matches the compressible reference data closely.

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The immersed boundary method case exhibits slightly decreased growth rates in comparison to both the body fitted DNS and reference compressible DNS. However, the results are acceptable, particularly in comparison to alternative immersed boundary methods tested for the same case by von Terzi et al. [24]. 4.3. Immersed boundary for trailing-edge serrations For the current study the immersed boundary method is used to represent the flat-plate extensions to the airfoil trailing edge illustrated in Fig. 1. The serrated and straight flat plates, both with same area, are intended to be thin; however, the immersed boundary method requires that the plates be at least two points thick in order to operate correctly. Since for the C-type domain grid points are duplicated on the wake dividing line, this means that the flat-plate extensions must be three grid points thick. This results in a flat plate that is 1.2  10  3 airfoil chords thick. In contrast to the sharp geometries investigated by Howe [8], for the current study both the root and tip of the trailing-edge serrations possess a finite radius of 3.5  10  3. Using a finite radius prevents image points from being located in regions of body points.  Our previous airfoil simulation at a ¼ 53 , Re =5  104 and M =0.4 yielded a displacement thickness of d ¼ 2:8  102 at the airfoil trailing edge, and hence by considering this in conjunction with Howe’s relationship oh=U 4 1, serration dimensions of spanwise length l ¼ 0:05 and streamwise length h= 0.03 were selected. This gives a length of 2h =0.06, which is of a similar order to the expected boundary layer thickness, and is predicted to result in noise reduction for frequencies f 4 5:3. Furthermore, the serration angle results in 393 and therefore less than 453 as required for noise reduction in Howe’s analysis. The spanwise domain width of z =0.2, as employed for previous DNS, means that the flow over four serrations is computed. To illustrate that the immersed-boundary implementation of the trailing-edge extension produces a no-slip condition with a smooth velocity profile, the time-averaged streamwise velocity profile is shown in Fig. 7(a). The first point with u o0 is the ghost point, located within the geometry, and the second point is the first grid point in the flow. The immersed boundary is located between the two points. The velocity profile is smooth with no indication of oscillations caused by the immersed boundary method. In addition, a top-view of the time-averaged pressure field at the first grid point above the serrated trailing-edge extension is shown in Fig. 7(b). This figure was produced from data where every fourth point in x and every second point in z were skipped. The serration tips are located at z =0, 0.05, 0.1 and 0.15. Although there is no grid-refinement in the spanwise direction, it appears as if the grid density is sufficiently high to cope with the geometric discontinuities in this direction. The presence of the serrations can be clearly identified in the pressure field, with pressure values lower between the serrations and peak values of pressure on the serrations. 5. Results The structure of the three-dimensional flowfield is shown for the straight and serrated trailing-edge cases in Fig. 8. Isosurfaces of Q illustrate the presence of hydrodynamic instabilities within the laminar region of the suction-side boundary layer. At the airfoil mid-chord, large structures, appearing ‘roller’-like, are observed to rapidly break down to small scales, and a turbulent boundary layer results. The only visible result in the flowfield between the two cases is a slightly stronger spanwise variation of the primary instability structures upstream of transition in the serrated case. A qualitative view of the flow in the vicinity of the trailing-edge serrations is shown in Fig. 9(a), showing the existence of horseshoe-type vortices generated by the serrations. The effect of these structures on the flow field in the vicinity of the trailing edge has been more thoroughly investigated in Jones and Sandberg [26]. 5.1. Effect on acoustic field Acoustic waves are illustrated by plotting iso-contours of rui in Fig. 8. All simulations exhibit greater intensity acoustic waves above the airfoil. This is because the transition/reattachment region is an important noise source for the current configuration [5], which dominates over trailing-edge noise at high frequencies. The acoustic radiation above the airfoil can therefore be considered to be the sum of the trailing-edge noise and the additional noise sources on the upper airfoil surface, whilst the acoustic radiation below the airfoil consists solely of trailing-edge noise. The acoustic waves below the airfoil are thus weaker in intensity and originate from the trailing edge. While it is difficult to detect any difference between the two cases for the acoustic field above the airfoil, acoustic waves below the airfoil appear to be slightly weaker for serrated trailing-edge cases than for the equivalent straight trailing-edge cases, implying that the trailing-edge noise contribution of the airfoil self-noise is affected by the serrations. All simulations were conducted until statistical convergence was achieved, allowing for analysis in the frequency domain. Power spectra of pressure from data of probe locations above and below the airfoil are shown in Fig. 9(b) for the straight and serrated trailing-edge cases. Above the airfoil, where acoustic waves generated both at the airfoil trailing edge and in the vicinity of transition are expected, the spectra appear in general to be similar in amplitude. Below the airfoil, where only trailing-edge noise is expected, for frequencies f 4 6 the serrated trailing-edge simulation exhibits reduced amplitude pressure fluctuations compared to the straight trailing-edge simulation. Contrary to Howe’s [8] analysis, the noise reduction does not occur for all frequencies above the threshold frequency, an effect also observed in experiments by Oerlemans et al. [9].

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0.12 0.1

y

0.08 0.06 0.04 0.02 0

0.2

0.4

0.6

0.8

1

u

Fig. 7. Time-averaged streamwise velocity profile on the trailing-edge extension for straight trailing-edge case (a); first point (with u o 0) is a ghost point; top-view of the time-averaged pressure field at the first grid point above the serrated trailing edge (b).

To visualize how the radiated sound field is modified by trailing-edge serrations, contours of modulus of pressure are shown in Fig. 10. In order to reduce the size of data stored to disk, the time series of pressure was stored only for the acoustic field, skipping every 4th point in x and every 2nd point in Z, and not the hydrodynamic field, which contains the majority of the total number of grid points. Therefore the hydrodynamic field is cut out. For clarity, the airfoil surface is denoted by a solid line. Three target frequencies were chosen, shown in Fig. 9(b) as vertical lines. One-third octave averaging about the target frequencies was performed to account for the broadband nature of the airfoil-noise. The lowest target frequency f= 3.37 is mainly due to TE noise because the spectra obtained above and below the airfoil are similar. More evidence for this conjecture is given in Jones et al. [5]. The mid-range frequency f=7.75 corresponds to the most amplified frequency of instabilities in the laminar-turbulent transition region. The structures in the laminar separated shear layer seen in Fig. 8 are consequences of these instabilities. This is, however, not to say that the transition process will necessarily act as a direct noise source. This particular frequency was selected as it is associated with an important hydrodynamic mechanism. Note that the acoustic feedback loop and the vortex shedding frequency for these flow conditions are considerably lower (f o 4), as reported in Jones et al. [10,5]. The highest frequency considered, f ¼ 11:2, was shown to be dominated by additional noise sources on the suction side [5], as also evidenced by the higher amplitudes of the pressure spectra obtained from above the airfoil. According to Howe [8], no TE noise reduction can be expected from serrations with the current dimensions at the lowest frequency chosen. This is confirmed in Figs. 10(a) and (d), where, considering the contours below the airfoil, no significant difference can be observed between the straight and serrated TE cases. The contrary is true for the higher frequencies shown. At f= 7.75, shown in Figs. 10(b) and (e), and f= 11.2, Figs. 10(c) and (f), the trailing-edge noise

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Fig. 8. Iso-contours of Q =100 coloured by streamwise vorticity over the range ox ¼ 7 100, with background showing dilatation rate over the range rui ¼ 7 7:5  102 , for case with straight (a) and serrated (b) trailing edge. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0.0001 1e-05

pp*

1e-06 1e-07 1e-08 1e-09 1e-10 1e-11

0

10

20 f

30

40

Fig. 9. Iso-surfaces of Q= 500, coloured by streamwise vorticity for levels [  100:100] for DNS of serrated TE (a). Power pressure spectra taken above and below the airfoil at (x,y)= (0.5, 7 0.5) for airfoils with straight (- - -) and serrated (—) trailing edges (b); vertical lines denote target frequencies and shaded areas show the range of frequencies used for one-third octave averages about the target frequencies. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

contribution appears considerably weakened by the addition of TE serrations. However, the addition of trailing-edge serrations also appears to considerably change the noise radiation on the suction side of the airfoil at the lowest frequency, adding an upstream pointing lobe not present in the straight trailing-edge case. This was unexpected, especially in light of

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Fig. 10. Logarithmically spaced contours of one-third octave averaged modulus of pressure for straight trailing edge at target frequencies f= 3.37 (a), 7.75 (b) and 11.2 (c), and serrated trailing edge at target frequencies f= 3.37 (d), 7.75 (e) and 11.2 (f).

the fact that statistical quantities of turbulence and surface pressure spectra on the suction side of the airfoil were shown in Jones and Sandberg [27] to be highly similar for both cases. The effect of the TE serrations on the hydrodynamic field is therefore investigated again. 5.2. Effect on hydrodynamic field An airfoil without trailing-edge extension at the same flow parameters as chosen here was shown in Jones et al. [10] to be subject to an acoustic feedback loop in which sound waves generated at the trailing edge excite the boundary layer upstream of separation. Thus the question arises whether TE serrations might possibly affect the laminar-turbulent transition process in case the sound waves scattered off the trailing edge display a non-zero spanwise wavenumber which is then forced upon the laminar separation bubble and potentially alter the convective instability mechanism. Indeed, shed vortices upstream of transition visualized by iso-surfaces of Q display a more significant spanwise variation for the serrated case, as shown in Fig. 8. The behaviour of the separation bubble is studied using probability density functions (PDFs) to account for its unsteadiness. Normalized PDFs of the skin friction coefficient cf on the suction surface of the airfoils are computed as described in Jones et al. [17]. Iso-contours of cf PDFs over the upper airfoil surface are plotted in Figs. 11(a) and (b) for straight and serrated trailing edges. The upper and lower PDF boundaries represent cf at three standard-deviations from the mean, hence where the PDF is very narrow cf varies only little with time, whereas where the PDF is wide cf varies strongly. The main observation that can be made is that the two cases, surprisingly, are almost identical. This implies that the variation of the separation bubble length is similar, although the PDFs do not reveal any information on whether the oscillation about the mean occurs at the same frequencies. Figs. 11(c) and (d) show the spanwise correlation function of pressure for the serrated and straight TE cases. It is difficult to identify any clear differences between the two cases. While the spanwise correlation is slightly decreased in the laminar-turbulent transition region, x 0:375, the correlation is slightly increased at the aft end of the airfoil for the serrated case. However, neither the PDF of the skin friction nor the spanwise correlation can detect small amplitude variations in the spanwise direction. Therefore, power spectra of the spanwise wavenumber of turbulent kinetic energy for the serrated and straight trailingedge cases are shown in Fig. 12. The spectra are integrated over the wall-normal direction at two streamwise positions, at x = 0.35, i.e. the laminar separated region, and x =0.99, just upstream of the trailing-edge extensions. In the transitional,

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Fig. 11. Contours of normalized probability density function of cf for straight trailing edge (a) and serrated trailing edge (b); contours of spanwise correlation function of pressure for straight trailing edge (c) and serrated trailing edge (d).

1e+00 1e-03

kk*

1e-06 1e-09 1e-12 1e-15 100 β Fig. 12. Power spectra of the spanwise wavenumber of turbulent kinetic energy for serrated (B) and straight (3) trailing-edge simulations. The spectra are integrated over the wall-normal direction at x= 0.35 (- - -) and x= 0.99 (—).

separated shear layer at x =0.35, the spectrum obtained from the simulation with a serrated trailing edge contains more energy for spanwise wavenumbers with b 4 30 than the spectra obtained from the straight case DNS, confirming the increased spanwise variation of the transitional structures observed in Fig. 8. Just upstream of the trailing-edge extension, at x= 0.99, the spectra from the serrated and straight trailing-edge cases look similar, indicating that the turbulent state at this location is not strongly affected by the upstream transition process. Overall, despite evidence that transitional structures exhibit greater spanwise variation where serrations are present (Figs. 8 and 12), it can be concluded from the current results that the presence of serrations does not significantly affect the overall hydrodynamic field, including the behaviour of the laminar separation bubble. Two reasons for why the hydrodynamic field is not considerably affected by the presence of serrations are suggested. Firstly, the feedback loop occurs at a frequency of f= 3.3 [28], which is below the threshold frequency of f=5.3 predicted by Howe [8] for which serrations are effective. Secondly, the three-dimensional instability mechanism reported in Jones et al. [17] dominates the transition process, thus even if the convective instability was forced differently it would not alter the transition process.

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The latter conjecture is supported by Fig. 12 which shows that spanwise spectra of turbulent kinetic energy are similar for serrated and straight trailing-edge cases close to the trailing edge. 6. Conclusion Direct numerical simulations of the flow around NACA-0012 airfoils at incidence were conducted with the aim of investigating the potential noise reduction of trailing-edge serrations and the effect of the serrations on an acoustic feedback loop. Power spectra of pressure recorded above and below the airfoil and one-third octave averaged contours of pressure show that trailing-edge noise is reduced at higher frequencies while no significant difference is observed for TE noise at low frequencies and for noise generated in the reattachment region. The negligible effect of the trailing-edge serrations on the hydrodynamic field is suggested to be firstly due to the fact that the feedback loop occurs at a frequency below the threshold frequency for which serrations are effective. Secondly, support was presented for the fact that the laminar-turbulent transition is dominated by a three-dimensional instability mechanism [17] which is unaffected by the serrations. In light of the hydrodynamic field being largely unaffected by the presence of trailing-edge serrations, it is suggested that the trailing-edge noise reduction is mainly due to the effect of the serrations upon the diffraction process. Currently, the differences between the two geometries in low frequency noise radiation above the airfoil is still unaccounted for. Finally, the feasibility of integrating an immersed boundary method, capable of representing arbitrary three-dimensional geometries, into a high-fidelity DNS solver for direct noise computations was demonstrated. This approach will enable DNS-based noise investigations of configurations previously not practicable, such as, e.g. surfaces containing tripping devices.

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