The Reynolds number of the current study is around 3 million. Although sound propagation is based on compressible air, the flow can still be considered as incompressible because the flow and acoustic solver can be split [13]. The filtered incompressible Navier-Stokes equations, momentum, turbulent stresses and eddy viscosity equations are applied to obtain the turbulent flow data. The filtered incompressible equations are solved by the in-house EllipSys3D code [16] [17]. The code is based on a multi-block/cell-centered finite volume discretization of the steady/unsteady incompressible Navier-Stokes equations in primitive variables (pressure and velocity). The EllipSys3D code is programmed using a multi-block topology, and therefore it is parallelized using a message-passing interface.

The turbulent stresses in Equation (1) is based on the mix model [18] [19] through Equations (2)-(4) where the first filter is identified with an over bar “-” which relates to the finest mesh. The second filter is denoted with a tilde “~” above. The second filter is also denoted as the test filter, it is computed on a mesh twice as coarse as the finest mesh. In Equation (3), ω is vorticity, Δ = (Δ_xΔ_yΔ_z)^1/3 is the reference filter size based on the grid volume. The model constants are C = 0.04 and α = 0.5.

In this work, the formulation proposed by Farassat [20] is applied. The formulation is the solution of the Lighthill’s acoustic analogy [11] with surface sources only when the surface moves at subsonic speed. This formulation has been successfully used for helicopter rotor and propeller noise predictions. At the retarded or emission time, the thickness and loading noise equations are written as

The right hand sides of Equations (5) and (6) are the integrations of time history variables obtained from flow calculations. The variables include wall normal velocity v_n, the Mach number of the source in the radiation direction M_r, the pressure on the solid wall surface p, the distance between source and receiver r, the angle between the radiation direction and the local wall normal direction θ. Figure 1 shows a sketch of an airfoil where dS indicates one of the typical wall element that is integrated over the entire airfoil surface. It is obvious that the the angle θ will contribute to the noise directivity. The acoustic solver may run in parallel with flow model, in practice the acoustic solver starts when the flow-field is fully established.

The wind tunnel used for the surface pressure measurements is optimized for aerodynamic performance. Since TE noise signal is difficult to obtain directly, the surface pressure measurements are performed to evaluate the noise source mechanisms [21]. The measurements are performed for a NACA 0015 airfoil, at a flow speed of 50 m/s, and the airfoil chord is 900 mm. The surface pressure microphones are identified with different numbers and mounted on the airfoil surface. The locations of the microphones are depicted in Figure 2. The microphones are placed along the chord direction, e.g., from leading edge to the trailing edge. They are also spread in the spanwise direction to minimize the flow disturbance due to the upstream microphones. Pressure signals from some of these measurement points (Marked with circles in Figure 2) will be compared with LES results.

A C-type mesh is created for numerical simulation. The mesh is block-structured with 48³ grid points in each block. The total number of blocks is 216 which contains a total number of grid points around 24 million. The first wall cell size is in the order of 10⁻⁵ chord and the ratio of ∆x/∆y is around 25 along the airfoil wall surface. An iso-vorticity plot is shown in Figure 3. At the angle of attack of 4˚, the flow separation is seen on the suction side surface. In Figure 4, the spectral density of the incompressible pressure is compared with measurement data. As seen from the figure, the agreements are found at frequencies below 1000 Hz. The deviation at higher frequencies is due to two reasons: the limited grid size used for LES filter; the wind tunnel fan noise causes high frequency noise, especially the tonal components.

An additional computation is carried out for the same airfoil with a serrated TE. The flow and mesh configurations are the same as the previous calculation. As seen from Figure 5, the extended length of the serration (from its root to tip) is 5% of the airfoil chord length and the span of the serration is 10% of the chord. The pressure signals obtained from original airfoil and from the serrated airfoil are compared. In Figure 6, the two signals are selected at x/c = 0.64 and x/c = 0.9. The curves with legend Org₁ and SE₁ are located at x/c = 0.9. The other two curves are plotted for location x/c = 0.64. As it is seen, the signal closer to TE contains more energy than the one in the upper stream. Also it is observed that near the TE region, the serrated airfoil has less spectral density amplitude which might indicate less noise generation.

The acoustic computation is performed when the turbulent flow is numerically stabilized. The integration of sound source is through Equation (6) where only the loading noise plays the role. The time history data of acoustic pressure is recorded during the simulation. After that, the Sound Pressure Level (SPL) of the original airfoil and the serrated airfoil is compared at a same receiver point which is located at 3-chord distance normal to the suction side of TE. It is interesting to see that the difference of the SPL is quite similar to the one we see in Figure 7. The strength of the incompressible pressure fluctuation at TE has the same behavior as the noise spectra. This indicates that the strength of pressure fluctuation at TE is responsible for noise generation. A small change of TE geometry might influence a lot of noise generation.