### 2.1 Wall-modeled large eddy simulation

The present study is based on a compressible finite volume solver [18]. The convection and viscous terms of the Navier–Stokes equations were both discretized with a second-order numerical scheme. A central scheme and an upwind scheme were combined to form the spatial scheme. The blending parameter was determined by the grid quality [18], which makes the scheme less dissipative in the good grid quality regions and more dissipative in the poor grid quality regions. The third-order Runge–Kutta method was used for time marching. Vreman’s [19] model was applied for subgrid stress closure.

The wall-stress model of the boundary layer, which is an important feature for reducing the computational cost, was implemented, as shown in the sketch in Fig. 1. The LES grid was quite coarse near the wall (usually the first grid layer Δ*y*+ > 10). A wall-model velocity profile was embedded at the bottom of the LES grid. The apex velocity of the wall-model profile was obtained from the LES grid at the interpolation point. Kawai et al. [20] suggested that the interpolation point should be in the log-layer, and there should be more than two LES cells under the interpolation point. A one-dimensional wall-model grid with high grid density was automatically generated to solve the wall-model equation.

In the wall model framework, the viscous effect was only considered in the *y*-direction. In the very near-wall region, the pressure was assumed to be independent of the inner layer, and a quasi-steady state was assumed in the bottom of the boundary layer; consequently, the boundary layer equation becomes a “stress balance model” [21] or an equilibrium model [12].

$$\frac{\partial }{\partial y}\left[\left(\nu +{\nu}_t\right)\frac{\partial {u}_i}{\partial y}\right]=0,\kern0.5em i=1,2,3$$

(1)

$${\displaystyle \begin{array}{cc}{\tau}_{w,i}={\left.\nu \frac{\partial {u}_i}{\partial y}\right|}_{y=0},& i=1,2,3\end{array}}$$

(2)

$${\nu}_t=\kappa y\sqrt{\frac{\tau_w}{\rho }}{\left[1-\exp \left(\frac{y^{+}}{A^{+}}\right)\right]}^2$$

(3)

$$\frac{\partial }{\partial y}\left[\left(\mu +{\mu}_t\right)u\frac{\partial u}{\partial y}+{c}_p\left(\frac{\mu }{Pr }+\frac{\mu_t}{{Pr}_t}\right)\frac{\partial T}{\partial y}\right]=0$$

(4)

As shown in Eq. (1), the equilibrium model is a one-dimensional ordinary differential equation (ODE). In the framework of a wall-modeled LES, the shear stress *τ*_{w}, as shown in Eq. (2), can be computed by the wall model and used as the wall boundary condition of the LES; consequently, Eq. (1) can be solved only for a resultant velocity because the shear stress components can be calculated by the velocity gradient and the normal vector of the wall. The eddy viscosity *ν*_{t} was estimated with a mixing length model, as shown in Eq. (3), where the constants *κ* and *A*^{+} were 0.41 and 17, respectively. The continuum equation is not affected by the wall model because of the impenetrability and the nonslip wall boundary condition. The energy equation of the wall model, as shown in Eq. (4), was used to solve the temperature distribution near the wall. The wall boundary condition for the LES energy equation was provided by the wall-model equation.

The wall boundary condition for Eq. (1) is a nonslip condition, and the apex velocity is interpolated from the instantaneous flow field of the LES. To solve the ODE, approximately 40 to 60 grid points with a stretching ratio of less than 1.05 are typically automatically generated. The Δ*y*+ value of the first wall-model cell is less than 1.0.

### 2.2 Ffowcs Williams–Hawkings equation

The far-field noise was assessed by the Ffowcs Williams-Hawkings (FW-H) equation [13]. Given a stationary sound source in a fluid moving at a constant speed **U** = (*U*_{1}, *U*_{2}, *U*_{3}), the convective FW-H equation for a permeable surface *S* is given by Eq. (5) [11, 22]. *Q*_{n}, *F*_{i} and *T*_{ij} are shown in Eq. (6). The subscript 0 denotes the free stream quantity, and *c*_{0} is the sound speed. The superscript ' represents the perturbation relative to the free stream; for example, *ρ*^{′} = *ρ* − *ρ*_{0}. According to the proposal of Spalart and Shur [3], the sound perturbation density *ρ*^{′} can be expressed as \({p}^{\prime }/{c}_0^2\). *H*(*S*) and *δ*(*S*) are the Heaviside function and Dirac delta function, respectively; \(\overset{\frown }{\mathbf{n}}=\left({\overset{\frown }{n}}_1,{\overset{\frown }{n}}_2,{\overset{\frown }{n}}_3\right)\) is the unit normal vector pointing outward from the surface *S*; *u*_{i} are the local flow velocities on *S*; *δ*_{ij} is the Kronecker delta; and *σ*_{ij} is the viscous stress tensor, which is often negligible in far-field sound computation.

$${\displaystyle \begin{array}{l}\left(\frac{\partial^2}{\partial {t}^2}+{U}_i{U}_j\frac{\partial^2}{\partial {x}_i\partial {x}_j}+2{U}_j\frac{\partial^2}{\partial t\partial {x}_j}-{c}_0^2\frac{\partial^2}{\partial {x}_j\partial {x}_j}\right)\left[{\rho}^{\prime }H(S)\right]\\ {}\begin{array}{cc}=\frac{\partial }{\partial t}\left[{Q}_n\delta (S)\right]-\frac{\partial }{\partial {x}_i}\left[{F}_i\delta (S)\right]+\frac{\partial^2}{\partial {x}_i\partial {x}_j}\left[{T}_{ij}H(S)\right],& i,j=1,2,3\end{array}\end{array}}$$

(5)

$${\displaystyle \begin{array}{l}{Q}_n=\left[\rho \left({u}_i+{U}_i\right)-{\rho}_0{U}_i\right]{\hat{n}}_i\\ {}{F}_i=\left[{P}_{ij}+\rho \left({u}_i-{U}_i\right)\left({u}_j+{U}_j\right)+{\rho}_0{U}_i{U}_j\right]{\hat{n}}_j\\ {}\begin{array}{cc}{T}_{ij}=\rho {u}_i{u}_j+{P}_{ij}-{c}_0^2{\rho}^{\prime }{\delta}_{ij},& {P}_{ij}={p}^{\prime }{\delta}_{ij}-{\sigma}_{ij}\end{array}\end{array}}$$

(6)

$${\displaystyle \begin{array}{cc}q\left(\omega \right)={\int}_{-\infty}^{+\infty }q(t)\exp \left(- i\omega t\right) d t,& q(t)=\frac{1}{2\pi }{\int}_{-\infty}^{+\infty }q\left(\omega \right)\exp \left( i\omega t\right) d\omega \end{array}}$$

(7)

$${\displaystyle \begin{array}{l}\left(-{\omega}^2+{U}_i{U}_j\frac{\partial^2}{\partial {x}_i\partial {x}_j}+2 i\omega {U}_j\frac{\partial }{\partial {x}_j}-{c}_0^2\frac{\partial^2}{\partial {x}_j\partial {x}_j}\right)\left[H(S){\rho}^{\prime}\left(\mathbf{x},\omega \right)\right]\\ {}= i\omega {Q}_n\left(\mathbf{x},\omega \right)\delta (S)-\frac{\partial }{\partial {x}_i}\left[{F}_i\left(\mathbf{x},\omega \right)\delta (S)\right]+\frac{\partial^2}{\partial {x}_i\partial {x}_j}\left[{T}_{ij}\left(\mathbf{x},\omega \right)H(S)\right]\end{array}}$$

(8)

By defining a Fourier transform pair as Eq. (7), Eq. (5) becomes Eq. (8). *ω* = 2*πf* is the angular frequency, where *f* is the frequency; *k* = *ω*/*c*_{0} is the wavenumber; and \(i=\sqrt{-1}\) is the complex number.

$$G\left(\mathbf{x},\mathbf{y}\right)=\frac{-1}{4\pi {R}^{\ast }}\exp \left(\frac{- i\omega R}{c_0}\right)$$

(9)

$${R}^{\ast }=\sqrt{{{\overline{x}}_1}^2+{\beta}^2\left({{\overline{x}}_2}^2+{{\overline{x}}_3}^2\right)}$$

(10)

$$R=\frac{R^{\ast }-{M}_0{\overline{x}}_1}{\beta^2}$$

(11)

$$\left.\begin{array}{l}{\overline{x}}_1=\left({x}_1-{y}_1\right)\cos \alpha \cos \phi +\left({x}_2-{y}_2\right)\sin \alpha +\left({x}_3-{y}_3\right)\cos \alpha \sin \phi \\ {}{\overline{x}}_2=-\left({x}_1-{y}_1\right)\sin \alpha \cos \phi +\left({x}_2-{y}_2\right)\cos \alpha +\left({x}_3-{y}_3\right)\sin \alpha \sin \phi \\ {}{\overline{x}}_3=-\left({x}_1-{y}_1\right)\sin \phi +\left({x}_3-{y}_3\right)\cos \phi \end{array}\right\}$$

(12)

By defining an observer location **x** = (*x*_{1}, *x*_{2}, *x*_{3}) and a source location **y** = (*y*_{1}, *y*_{2}, *y*_{3}), the three-dimensional free-space Green’s function for a convective wave equation can be expressed as Eq. (9). The definitions for *R*^{∗}, *R* and \({\overline{x}}_i,i=1,2,3\) are shown in Eqs. (10), (11), and (12), respectively. \({M}_0=\sqrt{U_1^2+{U}_2^2+{U}_3^2}/{c}_0\) is the Mach number. *α* and *ϕ* are the angle of attack and the sideslip angle, respectively, which are defined by \(\sin \alpha ={U}_2/\sqrt{U_1^2+{U}_2^2+{U}_3^2}\) and tan*ϕ* = *U*_{3}/*U*_{1}. \(\beta =\sqrt{1-{M}_0^2}\) is the Prandtl-Glauert factor.

With Green’s function, a solution for Eq. (8) in subsonic flow (*M*_{0} < 1) can be obtained, as shown in Eq. (13).

$${\displaystyle \begin{array}{l}4\pi {p}^{\prime}\left(\mathbf{x},\omega \right)H(S)=-{\int}_S i\omega {Q}_n\left(\mathbf{y},\omega \right)G\left(\mathbf{x},\mathbf{y}\right) dS\\ {}-{\int}_S{F}_i\left(\mathbf{y},\omega \right)\frac{\partial G\left(\mathbf{x},\mathbf{y}\right)}{\partial {y}_i} dS-{\int}_V{T}_{ij}\left(\mathbf{y},\omega \right)H(S)\frac{\partial^2G\left(\mathbf{x},\mathbf{y}\right)}{\partial {y}_i\partial {y}_j} dV\end{array}}$$

(13)

If *S* is a solid surface, then the source terms *Q*_{n} and *F*_{i} become Eq. (14). *T*_{ij} is the quadrupole term. If a permeable integration surface *S* contains a turbulent flow region, the quadrupole source must be considered. However, the quadrupole source is negligible for low-speed flow, as implied by the solid surface integration.

$${Q}_n=-{\rho}_0{U}_i{\overset{\frown }{n}}_i,\kern0.5em {F}_i=\left({P}_{ij}+{\rho}_0{U}_i{U}_j\right){\overset{\frown }{n}}_j$$

(14)