The governing equations used in the current study, termed the spatially-filtered Navier-Stokes equations, are obtained by applying a low pass filter on the time dependent Navier-Stokes equations in the physical space. The simulations were carried out using the commercial software ANSYS Fluent. The flow is assumed incompressible. In order to increase efficiency, the filter width is the same size as the mesh spacing used in the computational domain. The resulting equations describe the dynamics of large eddies [20] [21]. Field variables, such as pressure and velocity, are defined by their convolution with a filter function over the fluid domain, as following:

ϕ ¯ ( x ) = ∫ D ϕ ( x ′ ) G ( x , x ′ ) d x ′ (1)

where: D is the fluid domain and G is the filtering function. The overbar indicates spatial filtering and not temporal averaging. After applying the filter to the mass and momentum conservation equations, the NS equations become:

∂ ρ ∂ t + ∂ ∂ x i ( ρ u ¯ i ) = 0 (2)

∂ ∂ t ( ρ u ¯ i ) + ∂ ∂ x j ( ρ u ¯ i u ¯ j ) = ∂ ∂ x j ( σ i j ) − ∂ p ¯ ∂ x i − ∂ τ i j ∂ x j (3)

In order to obtain a close system of equations, the unknown SGS stresses are modeled by applying the Boussinesq eddy viscosity hypothesis [22], thus computing the sub-grid-scale turbulent stresses from:

τ i j − 1 3 τ k k δ i j = − 2 μ t S ¯ i j (4)

where: μ t is the subgrid-scale turbulent viscosity and τ k k is the isotropic part of the SGS. The latter part is not modeled as it is added to the filtered static pressure term. S i j is the strain-rate tensor of the resolved scale calculated from Equation (5) using the filtered velocity components.

S ¯ i j = 1 2 ( ∂ u ¯ i ∂ x j + ∂ u ¯ j ∂ x i ) (5)

Using the Wall Adapting Local Eddy Viscosity (WALE) model [23], μ t is modeled as:

μ t = ρ L s 2 ( S i j d S i j d ) 3 2 ( S ¯ i j S ¯ i j ) 5 2 ( S i j d S i j d ) 5 4 (6)

where: L s the mixing length of the sub-grid scale, and S i j d , which is a function of the strain and rotation rate tensors, are defined in Equations (7) and (8) as:

L s = min ( κ d , C w V 1 3 ) (7)

S i j d = 1 2 ( g ¯ i j 2 + g ¯ j i 2 ) − 1 3 δ i j g ¯ k k 2 (8)

with g i j defined as:

g ¯ i j = ∂ u ¯ i ∂ x j (9)

where: d is the distance to the closest wall, V is the volume of the computational cell, κ = 0.41 is the von Kármán constant, and C_w = 0.325 is the WALE constant. Values of d and V are determined by the ANSYS Fluent code for each mesh element. The von Kármán constant (defined as κ = y/l) is one of the few fundamental constants in fluid mechanics governing turbulent flows, and represents the ratio between mixing length, l, and the vertical distance in the flow field, y. Using an energy argument and mathematical symmetry it is argued that the von Kármán’s constant is κ = 0.414 (with derivable corrections for specific applications) [24]. As shown in [25], the best estimate for the von Kármán constant is found to be 0.40 ± 0.02. In the present simulation, κ = 0.41 was used, as specified in the ANSYS Fluent manual for external flows [20]. The WALE constant is used to estimate the mixing length of the subgrid scale, which is necessary to evaluate the unresolved (subgrid) turbulence of the flow in LES. Intensive validation during a European Union research project [23] has shown consistently superior results using C_W = 0.325 [26]. In addition, as described in the ANSYS Fluent manual [20], a value of the WALE constant of 0.325 has shown to yield satisfactory results for a wide range of flow; hence, this value was used in the current simulation.

In the present study and approach used, the entire domain is decomposed into clearly identifiable regions for RANS and LES before the simulation is started. This is usually referred to as segregated modeling. The goal is to use each model where it is best suited. The flow is initialized using RANS equations, which provide stationary field statistics, and LES re-solves the unsteady high-resolution perturbations near the TE, where it is needed. The main difficulty is defining proper interface conditions, seeing that inappropriate coupling could lead to results’ contamination in the LES or RANS subdomains. At the inflow interface, mass, momentum and energy are convected into the LES subdomain from the RANS region. The latter provides mean values which are to be coupled with the LES data. To obtain correct LES results, fluctuations must be provided at the interface and added to the mean flow computed by RANS. These fluctuations can be real, provided by precursor simulations or databases of similar flows, or synthetic, provided by Fourier modes, digital filters, random vortices, etc. The goal is to make the imposed fluctuation as close as possible to those present in a real physical flow.

The Vortex Method [27] was chosen as a means of adding artificial resolved turbulence at the RANS/LES interface. In this approach, a fluctuating vorticity field is added to the mean flow, consequently creating perturbations similar in behavior to realistic ones. The VM is based on the Biot-Savart law and the 2D evolution equation of vorticity. Vortex points, or particles, are distributed over the inlet interface perpendicular to the streamwise direction and are randomly convected, carrying information about the vorticity field. The amount of vorticity carried by a given particle “i” is represented by the circulation Γ according to Equation (11), and the assumed spatial distribution is given by Equation (12), such that:

ω ( x , t ) = ∑ i = 1 N Γ i ( t ) η ( | x − x i | , t ) (10)

Γ i ( x , y ) = 4 ( π A k ( x , y ) 3 N ( 2 ln ( 3 ) − 3 ln ( 2 ) ) ) (11)

η ( x ) = 1 2 π σ 2 ( 2 e − | x | 2 2 σ 2 − 1 ) 2 e − | x | 2 2 σ 2 (12)

where: N is the number of vortex points, A is the inlet section area, k is the turbulence kinetic energy and σ controls the size of the vortex particles. The resulting discretization for the velocity field is given by:

u ( x ) = 1 2 π ∑ i = 1 N     Γ i ( x i − x ) × z | x − x ′ i | 2 ( 1 − e | x − x ′ | 2 2 σ 2 ) e − | x − x ′ | 2 2 σ 2 (13)

in which z is a unit vector in the streamwise direction and x i is the location of the i-th vortex particle. The value of σ is calculated from a known profile of mean turbulence kinetic energy and mean dissipation rate at the inlet, such that:

σ = c k 3 2 2 ϵ (14)

in which c = Cμ^3/4 = 0.16, with Cμ = 0.09 as shown in [27]. Using the ANSYS Fluent code, the minimum value of σ is determined by the local mesh size to ensure that the vortices will always belong to the resolved scale. The sign of the circulation of each vortex is randomly changed every characteristic time scale, which is the time needed for a 2D vortex to travel n times its mean characteristic 2D size in the boundary normal direction, where n is set to equal 100 from numerical testing.

In order to overcome the prohibitive cost of directly resolving the pressure fluctuations responsible for noise in the far-field, a method based on Lighthill’s acoustic analogy [28] is used. In this approach, the nearfield flow is computed using the appropriate governing equations of ELES, and the far-field noise is predicted with the aid of an analytically de-rived integral solution to the wave equation. The acoustic analogy decouples sound generation from its propagation, thus allowing the separation of the flow solution from the acoustic analysis and the extraction of acoustic sources from the CFD domain.

The Ffowcs Williams and Hawkings (FW-H) formulation [29] adopts the most gen-eral form of Lighthill’s acoustic analogy. The FW-H equation [29] [30] is nothing but an in-homogeneous wave equation derived by manipulating the continuity and Navier-Stokes equation. The FW-H equation can be expressed as:

1 a ∞ 2 ∂ 2 p ′ ∂ t 2 − ∇ 2 p ′ = ∂ ∂ t { [ ρ ∞ v n + ρ ( u n − v n ) ] δ ( f ) }   − ∂ ∂ x i { [ P i j n j + ρ u i ( u n − v n ) ] δ ( f ) } + ∂ 2 ∂ x i ∂ x j { T i j H ( f ) } (15)

T i j = ρ u i u j + P i j − a ∞ 2 ( ρ − ρ ∞ ) δ i j (16)

where: p ′ = p − p ∞ is the sound pressure at the far-field, u i is the fluid velocity component in the x i direction, u n is the velocity component normal to the surface f = 0, v i is the surface velocity component in the x i direction, v n is the surface velocity component normal to the surface, δ ( f ) is the Dirac delta function and H ( f ) is the Heaviside function. The subscript “∞” denotes free-stream parameters. The f = 0 surface is a mathematical surface representing the source surface. n i is a unit vector normal pointing towards the exterior region of the source (f > 0), a ∞ is the speed of the sound at the far field, T i j is the Lighthill stress tensor defined in Equation (17), and P i j is the compressive stress tensor. The first term on the right hand side of Equation (16) represents the monopole or thickness source, modeling the sound generated by the displacement of a fluid as a body passes through it. The second term is the dipole or loading source, resulting from the unsteadiness of the forces acting on the body’s surface. The third term is the quadrupole source term, representing the non-linear fluctuations in the local sound speed and fluid velocity near the body surface. Monopole and dipole sources are dominant in low Mach number flows. By integrating Equation (16) assuming free-space flow and no obstacles between the sound source and receiver, a full solution consisting of surface and volume integrals is obtained. In the present case, the volume integral is neglected as it is only significant in high Mach number flows. Thus, far-field sound pressure can be expressed as:

p ′ ( x , t ) = p ′ T ( x , t ) + p ′ L ( x , t ) (17)

in which:

4 π p ′ T ( x , t ) = ∫ f = 0 [ ρ ∞ ( U ˙ n + U n ˙ ) r ( 1 − M r ) 2 ] d S     + ∫ f = 0 [ ρ ∞ U n { r M ˙ r + a ∞ ( M r − M 2 ) } r 2 ( 1 − M r ) 3 ] d S (18)

4 π p ′ L ( x , t ) = 1 a ∞ ∫ f = 0 [ L ˙ r r ( 1 − M r ) 2 ] d S + ∫ f = 0 [ L r − L M r 2 ( 1 − M r ) 2 ] d S     + 1 a ∞ ∫ f = 0 [ L r { r M ˙ r + a ∞ ( M r − M 2 ) } r 2 ( 1 − M r ) 3 ] d S (19)

U i = v i + ρ ρ ∞ ( u i − v i ) (20)

L i = P i j n ^ j + ρ u i ( u n − v n ) (21)

A dot over a variable indicates the source-time derivative of that variable, while the subscripts n, r and M denote the dot product with the unit normal vector, the unit radiation vector and surface velocity vector normalized by the speed of sound, respectively.

The airfoil selected for the present simulation is a NACA0012 symmetric airfoil, at zero angle of attack to isolate the effect of lift generation on the analysis. The chord length of the airfoils c is 0.3 m. The airfoil is placed in a square 10c × 10c domain, which was sufficient to provide convergence of results. The flow domain is divided into two regions as seen in Figure 1. RANS equations are employed in a coarse RANS domain, while LES equations are employed in a refined LES region near the TE. It’s important to note that only the noise radiated by the flow within the LES region is predicted in the numerical simulations. Since the presented work is focused on TE noise predictions, it is reasonable to neglect the noise generated by other airfoil sections, such as the leading edge. s is the span of the flow domain. Two embedded configurations were tested. For cases C1.1 and

C1.2, the LES domains in the streamwise direction extend from x c = 0.5 and x c = 0.7 , respectively, to 1c downstream of the TE. The letter “C’ stands for

computational. The origin is defined at the airfoil leading edge. In the transverse direction, the LES domain extends 0.25c above and below the airfoil. TE finlets are also tested. The thickness of the finlets, t, is 1 mm and the spacing between each two consecutive finlets, S, is 3.9 mm. The finlets do not extend past the airfoil

TE, and their LE falls at x c = 0.73 . The maximum height of the finlets (H), as measured from the airfoil surface, is 8 mm, and is reached at x c = 0.975 . The

design parameters are inspired by the work of Bodling et al. [12] and Shi et al. [31]. Figure 2 shows a general model of TE finlets. Table 1 summarizes the geometric parameters of the simulated cases. Computations are carried out at a free stream velocity u ∞ = 24   m / s and a free stream Mach number M ∞ = 0.071 ,

resulting in a chord-based Reynolds number, R e c = ρ u ∞ D μ , of approximately

500,000, where ρ is the fluid density, μ is the dynamic viscosity and D is the characteristic length, which is the airfoil chord in this case.

A predominantly hexahedral mesh is generated following the cartesian cut-cell method (Figure 3). This meshing technique, which has received a significant development in recent years [32], was found ideal for the current study seeing that it results in a smaller number of elements for the same resolution compared to other methods, thus significantly reducing the simulation time. In addition, the resulting elements are characterized by their high orthogonal quality and low skewness, which minimizes truncation errors [32] [33]. Element size is restricted to 25.6 mm in the coarse RANS zone, 0.8 mm in the refined LES zone, and 0.2 mm on the airfoil surface in the vicinity of the TE. Elements in the airfoil wake of the RANS zone have a size of 5 mm. The grid resolution in terms of wall-normal

units is defined by Δ x + = u τ Δ x ν , Δ y + = u τ Δ y ν and Δ z + = u τ Δ z ν where u τ is

the frictional velocity and ν is the kinematic viscosity. 40 inflation layers (Figure 3(c)) are generated around the airfoil with the thickness of the first layer set to 7.6 × 10⁻³ mm and a growth factor of 1.08, thus ensuring y + < 0.5 everywhere on the airfoil surface (Figure 4), at least three layers in the viscous sublayer and overall accurate boundary layer resolution. Table 2 lists mesh properties for all

simulated cases. The chosen computational grid has a maximum resolution Δ x max + ≤ 20 and Δ z max + ≤ 20 in the streamwise and spanwise directions, respectively [34] [35]. For verification, a third case was is simulated using two different meshes to investigate the effect of the mesh on the predicted noise. A steady-state mesh convergence study was carried out by progressively refining the mesh, creating three meshes having 6,666,668 elements, 7,606,083 elements and 9,011,531 elements, respectively. The values of integrated output parameters, such as lift and drag coefficients, were compared and the maximum error is found to be less than 0.4%, demonstrating mesh convergence. Furthermore, the first two meshes were carried over for a transient simulation analysis. The lift-history coefficients were evaluated for each mesh at every time step and their RMS values were computed. Both meshes yield the same lift-coefficient RMS value, c L RMS = 0.0013 . Consistent results in terms of integrated flow parameters for both steady-state and transient simulations, are a strong indication of the convergence of the used computational mesh, i.e., the mesh directly resolves enough flow structures for the results not to change with mesh refinement.

The boundary conditions used are demonstrated in Figure 5. A velocity inlet boundary condition is specified at the domain entrance, where u ∞ = 24   m / s . Periodic Boundary Conditions (PBCs) are applied on the right and left side walls of the domain in the spanwise direction to allow the flow to develop naturally. No-slip boundary conditions are applied on the airfoil surface and a zero gauge-pressure outlet boundary condition is used. The inlet turbulence is set to 0.3%. The SIMPLE pressure-velocity coupling scheme is used. All results are second order accurate in time and space. The flow is initialized using the k-ω SST model developed by Menter [36], as it gives accurate separation predictions for external flows. The VM is then used to inject turbulence at the RANS/LES interface and the simulation is run for 4T_TF “Through-flow time” to obtain a fully developed

flow, where T TF = L CFD u ∞ [37]. L_CFD is the LES domain length in the streamwise direction.

WALE Subgrid-Scale (SGS) modelling is employed in the LES region as it is designed to return correct asymptotic wall behavior for wall-bounded flows [20].

The time step d t = 1.2 × 10 − 5 seconds. With these values, the Courant-Friedrichs-Lewy (CFL) number achieved is ≤1 everywhere in the domain, meaning the flow particles don’t travel more than the length of one mesh element every time step. Residuals are reduced by three orders of magnitude each time step. Lastly, acoustic data is gathered for 3T_TF. All convergence residuals are set to 10⁻⁵. Pressure and velocity monitoring points were placed in the airfoil wake and statistical convergence is achieved. Statistical convergence is also achieved for the coefficients of lift and drag. All simulations are carried out using the commercial CFD software FLUENT 2019R3 and run on Intel Xeon L5410 2.33 GHz platform of 60 cores.

Experiments were conducted in the medium-speed, subsonic, closed-loop wind tunnel at Carleton University (Figure 6). A series of turbulence grids precede a 9:1 contraction, which reduces the turbulence intensity levels in the center of the test section to less than 0.27%. The tunnel has a removable, rectangular test section along with the surrounding anechoic chambers was completed to be used for aeroacoustic testing. This test section is a 0.78 m × 0.51 m rectangular section, 1.83 m long. The upper and lower walls of the test section are each composed of two aluminum sheet panels and contain hardware (circle aluminum material) for the vertical mounting of a two-dimensional airfoil in the midway, and 0.45 m from the upstream end of the test section [38] [39].

The airfoil wing is mounted vertically in the test section (Figure 7) with its leading edge (at zero AOA) 0.45 m downstream of the test section entrance. The airfoil under investigation is a NACA0012 airfoil with a finned TE serration, as shown in Figure 7. The chord length of the airfoil is 300 mm, and the width is

510 mm. Between the leading-edge ( x c = 0 ), and x c = 0.73   mm , the original

NACA0012 airfoil profile is unmodified, where x is the streamwise direction. Further downstream, 0.73 ≤ x c ≤ 1.0 , is a section that can be removed and replaced

by either an unmodified or modified TE profile. Once attached, the TE section forms a continuous profile giving the appearance that the serrations are cut into the main body of the NACA0012 airfoil. Typical parameters including the finlets height, H, and spacing, S, are defined as specified in Figure 2. The letter “E” stands for experimental. Far-field noise measurements in the mid-span were performed by a Brüel & Kjær microphone, which is installed at a distance of 1.4 m for an observer angle α = 90 ˚ . The analysis was carried out between 100 Hz and 5 kHz.

The pressure coefficient distribution around the airfoil is an important parameter since it influences the lift coefficient and the development of the boundary layer [40] [41]. In addition, the BL is responsible for the majority of the generated sound. C_p distributions for cases C1.1 and C1.2 are computed for validation and compared against experimental results obtained by Lee and Kang [42] for a NACA 0012 airfoil at a Re = 600,000, and full LES results published by Marsden, Bogey and Bailly [43] at R e c = 500000 (Figure 8). Excellent agreement is found between the computational and experimental surface pressure results. Of

importance is the fact that from x c = 0.15 down to the TE, the boundary layer

is subject to an adverse pressure gradient. Both cases C1.1 and C1.2 are validated against existing literature. The LES domain in C1.1 is longer in the streamwise

direction as it starts at a x c = 0.5 , while it starts at x c = 0.7 in C1.2 (see Table 1

and Table 2). Even though both configurations yield acceptable results, the configuration of case C1.2 is chosen for the succeeding simulations as the LES domain covers a larger span.

By calculating the coefficient of lift, c L = L 0.5 ∗ ρ v 2 A , for every timestep of flow

simulation, the lift-coefficient history can be plotted (Figure 9). L is defined as the lift force and A is the airfoil area. The lift-coefficient history is commonly used as an indicator of statistical convergence in transient simulations. Furthermore, it’s a non-dimensional representation of the fluctuating forces acting normal to the airfoil surface due to the turbulence of air flow. All axes are kept constant and aligned for the sake of clarity and comparison. For case C1.2, the lift monitor is random and irregular, characterized by a relatively small amplitude. The lift variation behavior in C2 is similar to C1.2. Figure 10 shows the instantaneous flow fields in the airfoil wake in term of iso-surfaces of the Q-criterion, which is defined as the second invariant of the instantaneous velocity gradient tensor [44]. The iso-surfaces are used to identify and portray the turbulent coherent structures of the wake, which are inherently three-dimensional. The iso-surfaces are colored by the spanwise vorticity, ω z , and demonstrate how the wake behavior changes as standard serrations are introduced then their geometrical parameters modified. For the case of a flat TE (C1.2), the wake is non-uniform and has almost no observable coherent structures. For C2, the wake looks similar and

no vortex shedding is observed. The effect of finlets on the instantaneous Turbulence Kinetic Energy (TKE) distribution is shown in Figure 11. Air is channeled between the finlets; the initially undisturbed boundary layer is deformed and decorrelated, and the highly energetic fluid particles are kept away from the airfoil surface, thus decreasing the efficiency of acoustic scattering [45] [46].

The FW-H aeroacoustic analogy [29] is used to compute the radiated far-field noise for the computational cases C1.2 and C2. In order to keep the computational cost reasonable, the span of the simulation domains is kept smaller than that of the experimental testing. Acoustic data is sampled every two flow-timesteps and data sampling is performed for 3T_TF after the flow is fully developed, resulting in a sampling frequency of 41.67 kHz and a frequency resolution of 28.4 Hz, where the frequency resolution is defined as the inverse of the sampling period. Pressure fluctuations are propagated to receivers placed midspan at a distance of

1.5 meters directly above the airfoils’ TEs. Discrete FFT is performed on the resulting time signals seen to compute the Sound Pressure Level (SPL) signal in the frequency domain, as shown in Figure 12. The Hanning window is applied to the time signal to reduce numerical leakages associated with the discrete FFT [47]. Both cases exhibit broadband behavior. No major discrepancies are seen in terms of general trend and no tonal peaks are present. A band-pass filter was applied obtained signals for C1.2 and C2 to calculate the overall sound pressure level (OASPL) for different frequency ranges, as defined in Equation (23). An OASPL reduction of 0.488 dB was predicted in the frequency range 1000 Hz to 5000 Hz.

As part of the current study, and in addition to the numerical predictions, wind tunnel testing has been performed to evaluate the noise of finlets at different angles of attack. The airfoil has a chord length c of 300 mm, and the width is similar to the width of the nozzle exit at 510 mm. The airfoil AOA is set to 0, 5, −5, 10 and −10 degrees and fixed to the nozzle exit by two side plates. The microphone was fixed and placed at about 1.4 m from the TE at a polar angle of 90˚ for all tests. The free jet velocity was set to 24 m/s and the flow parameters and

chord length are similar to the computational cases, yielding a Re_c of approximately 500,000. The fluctuating pressure-time signals for the used microphone are recorded and then used to calculate the SPL spectrum. The data sampling frequency is set to 20 kHz and the data sampling period is 30 seconds, corresponding to a frequency resolution of 0.033 Hz. The obtained signal is also passed through a time-domain filter to remove the low and high frequency contamination, caused by the microphone’s low frequency roll off and high-frequency aliasing. The band-pass filter used is a Butterworth filter with the first and second

stopband frequencies of 100 and f s 2 Hz respectively, where f s is the sampling

frequency. The sound pressure level, SPL, is computed using the root mean square (RMS) of filtered pressure signal as following:

SPL = 10 log 10 ( P R M S 2 P r e f 2 ) (22)

where: P_RMS is the root mean square pressure, and P_ref is the standard reference pressure in air, 20 μPa.

In Figure 13 the obtained SPL levels are displayed for AOA = 0˚, 5˚, −5˚, 10˚ and −10˚. For all the presented cases, the introduction of finlets did not alter the general shape of the acoustic spectra. The acoustic spectra kept the same broadband behavior, with slight differences in SPL amplitudes. The difference in amplitude between the two configurations is sensitive to the AOA, and is most pronounced in AOA = ±10˚. Finlets did not affect the broadband behavior of the radiated noise and no tonal peaks are observed. The OASPL is calculated to better quantify the effect of finlets on the radiated noise. Table 3 shows the obtained noise reduction as a function of the airfoil AOAs, where:

ΔOASPL = OASPL_ ( finnedTE ) − OASPL_ ( straightTE ) (23)

It is shown that noise reduction increases as the AOA is increased, and that values are consistent for the same absolute angles (ΔOASPL_−5˚ ≈ ΔOASPL_5˚ and ΔOASPL_−10˚ ≈ ΔOASPL_10˚). Since noise reduction is a function of the AOA, and modifying the AOA changes the BL thickness, then the noise reduction efficiency of finlets is dependent on the BL thickness. The observed trend agrees with

the findings of Gstrein et al. [48]. Experimental results confirm that finlets reduce the generated broadband noise. Even tough the overall noise reduction is not significant (as shown in Table 3), it is important to note that this solution can potentially become attractive to systems that contain several or extensive trailing edge structures or components, e.g., an aircraft wing or lifting surface, and a wind turbine.