3.1. Numerical set-up

To investigate the space–time correlations and spectral characteristics of wall pressure fluctuations, we use the wall-resolved large-eddy simulation (LES) method to simulate a fully-developed turbulent boundary layer over a flat plate. Considering that in many applications where TE noise is important the Mach number is relatively low, we choose to perform an incompressible simulation. Compared to the direct numerical simulation (DNS) method, LES require fewer grid points for wall-resolved simulations (Schlatter et al. Reference Schlatter, Li, Brethouwer, Johansson and Henningson2010), leading to less computational resource demands. The LES method solves the spatially filtered Navier–Stokes equations using a subgrid-scale (SGS) model. For incompressible flows, the filtered momentum and continuity equations can be expressed as

(3.1)\begin{gather} \dfrac{\partial \bar{u}_i}{\partial t}+\dfrac{\partial}{\partial x_j}(\bar{u}_i\bar{u}_j)=-\dfrac{1}{\rho_0}\,\dfrac{\partial\bar{p}}{\partial x_i}+\nu\,\dfrac{\partial^2\bar{u}_i}{\partial x_j\,\partial x_j}-\dfrac{\partial \tau_{ij}}{\partial x_j},\quad i=1,2,3, \end{gather}

(3.2)\begin{gather} \dfrac{\partial \bar{u}_j}{\partial x_j}=0, \end{gather}

where the overbar denotes filtered variables with a filter width $\Delta$, $t$ is the time, $u_i$ is the velocity component in the $x_i$-direction (also denoted as $u$, $v$ or $w$), $\rho _0$ is the density, $p$ is the pressure, and $\nu$ is the kinematic viscosity. The contributions from SGS components are represented through the SGS stresses $\tau _{ij}=\overline {u_iu_j}-\bar {u}_i\bar {u}_j$, which need to be modelled. Following the eddy-viscosity assumption, the SGS stress can be modelled as

(3.3)\begin{equation} \tau_{ij}-\tfrac{1}{3}\,\delta_{ij}\tau_{kk}=-2\nu_T\bar{{\mathsf{S}}}_{ij}, \end{equation}

where $\delta _{ij}$ is the Kronecker delta, and $\bar {{\mathsf{S}}}_{ij} = (\partial \bar {u}_i/\partial x_j+\partial \bar {u}_j/\partial x_i)/2$ is the large-scale strain-rate tensor. The SGS viscosity $\nu _{T}$ can be calculated using various models. In this study, the wall-adapting local eddy-viscosity (WALE) model (Nicoud & Ducros Reference Nicoud and Ducros1999) is used due to its ability to account for the wall effect on the turbulent structure. The value of $\nu _T$ is obtained as

(3.4) \begin{equation} \nu_T =C_w^2\varDelta^2\,\dfrac{({\mathsf{S}}_{ij}^d{\mathsf{S}}_{ij}^d)^{{3}/{2}}}{(\bar{{\mathsf{S}}}_{ij}\bar{{\mathsf{S}}}_{ij})^{{5}/{2}}+({\mathsf{S}}_{ij}^d{\mathsf{S}}_{ij}^d)^{{5}/{4}}}, \end{equation}

where $C_w$ is a constant coefficient, and ${\mathsf{S}}_{ij}^d$ is the traceless symmetric part of the square of the velocity gradient tensor,

(3.5)\begin{equation} {\mathsf{S}}_{ij}^d=\tfrac{1}{2}\left(\bar{{\mathsf{g}}}_{ij}^2+\bar{{\mathsf{g}}}_{ji}^2\right)-\tfrac{1}{3}\delta_{ij}\bar{{\mathsf{g}}}_{kk}^2. \end{equation}

Here, $\bar {{\mathsf{g}}}_{ij}={\partial \bar {u}_i}/{\partial x_j}$.

For the turbulent inlet boundary condition, we use the synthetic turbulent inflow generator. The generator employs a 2-D filter to produce spatially correlated 2-D slices of data. The instantaneous velocity on the slice is computed as

(3.6)\begin{equation} u_i=\bar{u}_i+{\mathsf{a}}_{ij}\varPsi_j, \end{equation}

where $\varPsi _j$ denotes the filtered fluctuating velocity field, and ${\mathsf{a}}_{ij}$ is the amplitude tensor, which is related to the Reynolds stresses tensor ${\mathsf{R}}_{ij}$ by

(3.7) \begin{equation} {\mathsf{a}}_{ij} = \begin{bmatrix} \sqrt{{\mathsf{R}}_{11}} & 0 & 0\\ \dfrac{{\mathsf{R}}_{21}}{{\mathsf{a}}_{11}} & \sqrt{{\mathsf{R}}_{22}-{\mathsf{a}}_{21}^2} & 0\\ \dfrac{{\mathsf{R}}_{31}}{{\mathsf{a}}_{11}} & \dfrac{{\mathsf{R}}_{32}-{\mathsf{a}}_{22}{\mathsf{a}}_{31}}{{\mathsf{a}}_{22}} & \sqrt{{\mathsf{R}}_{33}-{\mathsf{a}}_{31}^2-{\mathsf{a}}_{32}^2}\\ \end{bmatrix}. \end{equation}

By applying spatial and temporal filters to the random array sequences, we can introduce the desired temporal and spatial correlations in the instantaneous velocity fluctuations. The spatial turbulent length scale is defined by the two-point correlation, which is given by

(3.8)\begin{equation} L_i^j(\boldsymbol{x})=\int_0^\infty\dfrac{\overline{u_i'(\boldsymbol{x})\,u_i'(\boldsymbol{x}+\boldsymbol{e}_jr)}}{\overline{u_i'(\boldsymbol{x})\,u_i'(\boldsymbol{x})}}\,\mathrm{d}r, \end{equation}

where $r$ is the spatial separation in the $j$-direction, and $u_i'(\boldsymbol {x})$ denotes the velocity fluctuations. The parameters required to generate a turbulent inlet condition using the digital filter method (DFM) include the profiles of the mean velocity, turbulent Reynolds stresses and turbulent length scales. These parameters can be obtained through various approaches, such as precursor DNS, modelling from a Reynolds-averaged Navier–Stokes computation, or measurements from experiments. In this work, the mean velocity and turbulent Reynolds stresses are obtained from the DNS data provided by Schlatter & Örlü (Reference Schlatter and Örlü2010). The Reynolds number at the inlet, based on the momentum thickness $\theta$ and the free-stream velocity $U_0$, is set to 1410. Regarding the turbulent length scale $L_u^x$, a constant value may be prescribed. Following Wang et al. (Reference Wang, Vita, Fraga, Lyu, Wang and Hemida2022), $L_u^x$ is set to the boundary layer thickness scaled by a factor 0.6, and other turbulent length scales can be prescribed based on $L_u^x$. Table 1 shows the nine turbulent length scales used at the inlet with the DFM, where $\delta _{in}$ denotes the boundary layer thickness at the inlet.

Table 1. Turbulent length scales used in the inlet boundary condition.

The numerical simulation in this study is conducted using OpenFOAM-v2206. The computational domain, as illustrated in figure 1, has dimensions $50\delta _{in}\times 3.3\delta _{in}\times 3\delta _{in}$ in the streamwise ($x$), wall-normal ($y$), and spanwise ($z$) directions, respectively. The mesh cells are distributed exponentially along the $y$-axis and placed uniformly along the streamwise and spanwise directions. Table 2 lists detailed parameters employed in this study. To ensure grid independence, both fine and coarse meshes are tested, and the results of the grid independence test can be found in Appendix A.

Figure 1. Computational domain of the LES and an instantaneous flow field from the inlet to $19\delta _{in}$ downstream. The flow is visualized using the Q-criterion ($Q=2.5\times 10^4$) and coloured by the streamwise velocity (with levels increasing, the colour changes from blue to red).

Table 2. Grid information and free-stream velocity for the LES of a spatially developing turbulent boundary layer.

For the velocity boundary condition, a slip condition is used on the top wall, while a no-slip condition is imposed on the bottom wall. In the lateral direction, a periodic boundary condition is employed to simulate an infinite domain. At the outlet, the inletOutlet boundary condition is imposed. Regarding the pressure boundary conditions, all boundaries are set to zero gradient except for the top boundary, where a fixed pressure is prescribed.

3.2. Numerical results

In this subsection, we present the simulation results of a spatially developing turbulent boundary layer using the LES method. The turbulent statistics are obtained after the flow field reaches a statistically stationary state. The LES data are used to show the spatial evolution of the flow structures, validate the flow statistics, and examine the statistical characteristics of wall pressure fluctuations.

3.2.1. Flow field

An instantaneous flow field is visualized using an isosurface of the Q-criterion, as shown in figure 1. The computational domain is sufficiently long in the streamwise direction, and the flow region from the inlet to $19\delta _{in}$ downstream is selected for visualization. It can be seen that the unsteady flow structures generated at the inlet are not physical. But these unphysical structures quickly decay and the flow becomes more physical after approximately $10 \delta _{in}$. Similar phenomena can be seen in figure 2, where instantaneous snapshots of the flow field are displayed. Figure 2(a) shows the side view of the instantaneous streamwise velocity field in the $x$–$y$ plane, while figure 2(b) shows the overview in the $x$–$z$ plane located at $y=0.3\delta _{in}$. The visualization demonstrates that artificial turbulent structures are instigated at the inlet, preserved for a short distance downstream, and subsequently replaced by more physical structures.

Figure 2. Instantaneous snapshots of the streamwise velocity: (a) side view in the $x$–$y$ plane, and (b) overview in the $x$–$z$ plane at $y=0.3\delta _{in}$.

Flow statistics are obtained by averaging over the spanwise direction $z$ and time $t$. Therefore, the streamwise velocity can be decomposed into $u=U+u^\prime$, where $U$ and $u^\prime$ denote the mean velocity and the fluctuating velocity, respectively. The wall shear stress can be calculated as $\tau _w=\mu (\mathrm {d}U/\mathrm {d} y)|_{y=0}$, where $\mu$ is the dynamic viscosity. The friction velocity is defined as $u_{\tau }=\sqrt {\tau _w/\rho _0}$, and the characteristic length is given by $l_{\star }=\nu /u_{\tau }$. Therefore, the mean velocity and distance normal to the wall can be expressed in non-dimensional forms as $U^+=U/u_{\tau }$ and $y^+=y/l_{\star }$, respectively. In the subsequent analysis, the streamwise location $x=40\delta _{in}$ is used as the reference position. Table 3 provides the parameters of the turbulent boundary layer at this position.

Table 3. Turbulent boundary layer parameters at the streamwise location $x=40\delta _{in}$.

Figure 3 shows the distributions of the mean velocity and turbulent Reynolds stress components. The simulated mean velocity exhibits good agreement with values obtained by Wang et al. (Reference Wang, Vita, Fraga, Lyu, Wang and Hemida2022) using the dynamic Smagorinsky model, as shown in figure 3(a). In the near-wall region, the simulated mean velocity profile collapses well with the linear law. However, in the logarithmic region, both the simulation results of the present study and Wang et al. (Reference Wang, Vita, Fraga, Lyu, Wang and Hemida2022) slightly deviate from the log law. From figure 3(b), we can see that the simulated velocity fluctuations and Reynolds shear stress are in good agreement with DNS results. Slight deviations from the DNS profile can be seen for the ${u^{\prime +}_{rms}}$ profile in both the present study and the work of Wang et al. (Reference Wang, Vita, Fraga, Lyu, Wang and Hemida2022), which might be attributed to the artificial inflow-boundary condition employed in these two works. Nevertheless, figure 3 shows that the present LES capture essential flow physics.

Figure 3. Profiles of mean flow statistics at $x=40\delta _{in}$: (a) velocity and (b) Reynolds shear stress as well as streamwise, spanwise and wall-normal velocity fluctuations.

3.2.2. Properties of wall pressure fluctuations

In this part, we examine the statistical characteristics of wall pressure fluctuations. The key properties for noise predictions such as the frequency-dependent correlation lengths are presented here, and other properties, such as space–time correlations, can be found in Appendix B.

Figure 4 shows the single-point spectrum of the pressure fluctuations obtained using Welch's method (Welch Reference Welch1967) and normalized by the dynamic pressure $q_{\infty }=\rho _0U_0^2/2$ and the displacement thickness $\delta ^\ast$. The simulated pressure spectrum exhibits two distinct regimes: a $-1$ scaling regime, and a $-5$ scaling regime. The $-1$ scaling is associated with the eddies present in the logarithmic region of the boundary layer. These eddies contribute to the energy distribution in the low-frequency range of the spectrum. On the other hand, the $-5$ scaling, appeared in the high-frequency range, is related to the presence of smaller-scale eddies within the buffer layer (Blake Reference Blake2012).

Figure 4. Single-point spectrum scaled with outer parameters.

The mean convection velocity $\bar {U}_c$ can be estimated by determining the time delay $\tau$ corresponding to the correlation peak shown in figure 25 of Appendix B for a fixed spatial separation $\xi$, i.e.

(3.9)\begin{equation} \bar{U}_c(\xi)=\dfrac{\xi}{\tau}. \end{equation}

On the other hand, to obtain a frequency-dependent convection velocity, we can use the cross-spectral density $G_{pp}(\xi,\eta,\omega )$, which is a complex function (Gloerfelt & Berland Reference Gloerfelt and Berland2013). Let $\theta _p(\xi,\omega )$ denotes the phase of $G_{pp}(\xi,0,\omega )$. Then the phase velocity $U_{cp}(\xi,\omega )$ can be determined by (Farabee & Casarella Reference Farabee and Casarella1991)

(3.10)\begin{equation} U_{cp}(\xi,\omega)=-\dfrac{\omega \xi}{\theta_p(\xi,\omega)}. \end{equation}

The cross-spectral density can be written as

(3.11)\begin{equation} G_{pp}(\xi,\eta,\omega)=|G_{pp}(\xi,\eta,\omega)|\,\mathrm{e}^{-\mathrm{i}\xi\omega/U_{cp}(\xi,\omega)}. \end{equation}

The exponential term in (3.11) represents the convection behaviour of turbulent eddies, where the phase velocity is expected to be dependent on the frequency $\omega$ and the separation distance $\xi$.

Figure 5(a) shows the comparison between the simulated mean convection velocity and the experimental measurements by Bull (Reference Bull1967). It can be seen that there is good agreement between the two. As the streamwise separation increases, the mean convection velocity also increases and approaches 0.8. Note that most serration amplitudes are within $10\unicode{x2013}20\delta ^\ast$. The variation of the mean convection velocity $\bar {U}_c$ is very small in these distances. Figure 5(b) shows the variations of the phase velocity as functions of frequency for various fixed streamwise separations. The convection velocities calculated using the empirical models proposed by Smol'yakov (Reference Smol'yakov2006) and Bies (Reference Bies1966) (see Appendix C) are also shown for comparison. It is evident that the phase velocity increases with increasing streamwise separation, and as the frequency increases, the phase velocity rises rapidly, reaches a peak velocity, and then decays slowly. This suggests that the assumption of frozen turbulence, which assumes that all eddies in the turbulent boundary layer convect at the same velocity, is not strictly valid (Farabee & Casarella Reference Farabee and Casarella1991). Both the Smol'yakov model and the Bies model agree with the numerical results at intermediate and high frequencies, but the Bies model fails to capture the characteristics at low frequencies.

Figure 5. (a) Comparison of the mean convection velocity. (b) Phase velocity as a function of frequency for fixed streamwise separations with increment $1.6\delta ^\ast$ as well as convection velocities calculated using the Smol'yakov model and the Bies model.

Figure 6 examines the streamwise and spanwise coherences of pressure fluctuations. We see that the contour shapes of the coherence functions in the two directions are similar. For a fixed non-zero separation, the coherence increases with the increase of frequency and then decays. This behaviour indicates that the low-frequency components, associated with large-scale structures, maintain their coherence over longer distances, while the high-frequency components lose their coherence more rapidly with increasing separation. In particular, for the non-dimensional frequency $\omega \delta ^\ast /U_0>1$, the wall pressure fluctuations quickly lose their coherence as the separation increases, indicating that the perfect coherence assumed by the frozen turbulence might lead to significant errors. The coherence contours provide valuable information for determining the frequency-dependent correlation lengths.

Figure 6. (a) Streamwise and (b) spanwise coherences of wall pressure fluctuations.

Equations (2.8) and (2.9) provide the definitions of the streamwise and spanwise frequency-dependent correlation lengths. However, in practical applications, curve-fitting approaches are often employed. For each discrete frequency, the frequency-dependent correlation lengths can be assumed in exponential forms, i.e.

(3.12)\begin{gather} \gamma(\xi,0,\omega)=\mathrm{e}^{-|\xi|/l_x(\omega)}, \end{gather}

(3.13)\begin{gather} \gamma(0,\eta,\omega)=\mathrm{e}^{-|\eta|/l_z(\omega)}. \end{gather}

In figure 7, the frequency-dependent correlation lengths in the streamwise and spanwise directions are plotted as functions of frequency. Three empirical models, i.e. the Corcos model (Corcos Reference Corcos1964), the Smol'yakov model (Smol'yakov Reference Smol'yakov2006), and the Hu model (Hu Reference Hu2021) are also shown for comparisons, whose formulations can be found in Appendix D. It can be seen that both correlation lengths increase slightly as frequency increases in the low-frequency range. When the frequency further increases, $l_x(\omega )$ and $l_z(\omega )$ decay rapidly. All three empirical models exhibit similar decay trends within the intermediate- and high-frequency ranges. However, the Smol'yakov model and the Hu model could capture the characteristics at low frequencies. In addition, at higher frequencies, we can see that the simulated correlation lengths decay slowly and even increase. This phenomenon can also be found in the study of Van Der Velden et al. (Reference Van Der Velden, Van Zuijlen, De Jong and Bijl2015), and a mesh refinement may be helpful to obtain improved decay tendencies of the frequency-dependent correlation lengths in this regime.

Figure 7. Frequency-dependent correlation lengths in (a) the streamwise direction and (b) spanwise direction.

An interesting observation is that for the same frequency, $l_x(\omega )>l_z(\omega )$. This is in contrast to the streamwise frequency-independent correlation length $\varLambda _x$, which is smaller than the spanwise correlation length $\varLambda _z$. This phenomenon can be attributed to the convection of turbulence in the streamwise direction. Considering the definition of the frequency-dependent correlation length, we have

(3.14)\begin{equation} l_x(\omega)=\int_{0}^{\infty}\dfrac{|G_{pp}(\xi,0,\omega)|}{\phi(\omega)}\,\mathrm{d}\xi. \end{equation}

From (3.14), we can see that $l_x(\omega )$ characterizes the correlation that eliminates the effect of the streamwise convection of turbulent eddies. Physically, this represents the correlation length measured in a coordinate frame that moves with the eddy.

Introducing the complex form of the cross-spectral density, the space–time correlation in the streamwise direction can be written as

(3.15)\begin{equation} Q_{pp}(\xi,0,\tau)=\int_{-\infty}^{\infty}|G_{pp}(\xi,0,\omega)|\exp({\mathrm{i\omega}(\tau-\xi/U_{cp}(\xi,\omega))})\,\mathrm{d}\omega. \end{equation}

Therefore, the frequency-independent correlation length reads

(3.16)\begin{equation} \varLambda_x=\dfrac{1}{Q_{pp}(0,0,0)}\int_{-\infty}^{\infty} \left|\int_{-\infty}^{\infty}|G_{pp}(\xi,0,\omega)|\exp({-\mathrm{i\omega}\xi/U_{cp}(\xi,\omega)})\,\mathrm{d}\omega\right|\mathrm{d}{\xi}. \end{equation}

Comparing (3.14) and (3.16), it is clear that the calculation of the streamwise frequency-independent correlation length $\varLambda _x$ takes into account the influence of the convection of turbulent structures. On the other hand, the frequency-dependent correlation length $l_x(\omega )$ does not. Since the convection of eddies contributes to the decay of the correlation, it is possible that $\varLambda _x < \varLambda _z$ even though $l_x(\omega )>l_y(\omega )$.

In this part, we have examined two important features of wall pressure fluctuations that are omitted by the frozen turbulence assumption. First, the convection velocity is not strictly constant. Second, the eddies lose their coherence as they convect downstream, leading to a finite streamwise correlation length. These two characteristics are important non-frozen properties and must be accounted for in the modelling of the far-field noise emitted from serrated trailing edges.

4.1. Model establishment

With the properties of wall pressure fluctuations and the numerical results discussed above, we are in a position to consider the influence of non-frozen turbulence on the noise prediction for serrated trailing edges. As shown in figure 8, consider a flat plate encountering a uniform flow. The plate has chord length $c$, span $d$, and a trailing edge with serrations of amplitude $2h$ and wavelength $\lambda$.

Figure 8. Schematic of a flat plate with TE serrations.

According to Lyu et al. (Reference Lyu, Azarpeyvand and Sinayoko2016), for a general wall pressure fluctuation characterized by its wavenumber–frequency spectrum $\varPi (k_1, k_2, \omega )$, the far-field acoustic power spectral density $S_{pp}$ at the observer position $\boldsymbol {X}_0=(X_0,Y_0,Z_0)$ is found to be

(4.1)\begin{align} S_{pp}(\boldsymbol{X}_0,\omega)=\left(\dfrac{\omega Y_0c}{4{\rm \pi} c_0S_0^2}\right)^22{\rm \pi} d\sum_{m=-\infty}^{\infty}\int_{-\infty}^{\infty}|\mathcal{L}(k_1,2m{\rm \pi}/\lambda,\omega)|^2\,\varPi(k_1,2m{\rm \pi}/\lambda,\omega)\,\mathrm{d}k_1, \end{align}

where $c_0$ is the speed of sound, $S_0^2=X_0^2+(1-M_0^2)(Y_0^2+Z_0^2)$, $M_0$ is the Mach number of the flow, and $\mathcal {L}$ is the response function, which could be calculated iteratively. Note that there is no assumption regarding the frozen property of the wall pressure fluctuations here. More details about this model can be found in Lyu et al. (Reference Lyu, Azarpeyvand and Sinayoko2016).

It can be seen from (4.1) that the formulation relies on the wavenumber–frequency spectrum as input, and involves an infinite integral over the streamwise wavenumber $k_1$. While this integral can be evaluated numerically, such an approach would be computationally demanding. Furthermore, obtaining an accurate wavenumber–frequency spectrum $\varPi (k_1,k_2,\omega )$ is challenging in both numerical simulations and experimental measurements. To achieve a convenient prediction and determine the physical impact of non-frozen turbulence on TE noise, we aim to develop a simplified model based on the characteristics of wall pressure fluctuations. Specifically, we approximate the cross-spectral density using a variable-separation form:

(4.2)\begin{equation} G_{pp}(\xi,\eta,\omega)=\gamma(\xi,0,\omega)\,\mathrm{e}^{-\mathrm{i}\xi\omega/U_c(\omega)}\,G_{pp}(0,\eta,\omega). \end{equation}

This form is similar to the Corcos model (Corcos Reference Corcos1964), which was initially developed by fitting experimental data and has since been widely used in the modelling of wall pressure fluctuations. However, the Corcos model is more stringent as it assumes that the normalized cross-spectral density can be represented by a function that depends on a single dimensionless variable. Here, the convection velocity is expressed as a function of the angular frequency, while its dependency on the streamwise separation is neglected for simplicity.

By performing Fourier transforms, the wavenumber–frequency spectrum can be written as

(4.3)\begin{align} \varPi(k_1,k_2,\omega)&=\dfrac{1}{(2{\rm \pi})^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\gamma(\xi,0,\omega)\,\mathrm{e}^{-\mathrm{i}\xi\omega/U_c(\omega)}\,G_{pp}(0,\eta,\omega)\,\mathrm{e}^{\mathrm{i}k_1\xi}\,\mathrm{e}^{\mathrm{i}k_2\eta}\,\mathrm{d}\xi\,\mathrm{d}\eta\nonumber\\ &=\phi_x(k_1,\omega)\,\phi_z(k_2,\omega), \end{align}

where

(4.4)\begin{equation} \phi_x(k_1,\omega) = \dfrac{1}{2{\rm \pi}}\int_{-\infty}^{\infty}\gamma(\xi,0,\omega)\,\mathrm{e}^{-\mathrm{i}\xi\omega/U_c(\omega)}\,\mathrm{e}^{\mathrm{i}k_1\xi}\,\mathrm{d}\xi \end{equation}

and

(4.5)\begin{equation} \phi_z(k_2,\omega)=\dfrac{1}{2{\rm \pi}}\int_{-\infty}^{\infty}G_{pp}(0,\eta,\omega)\,\mathrm{e}^{\mathrm{i}k_2\eta}\,\mathrm{d}\eta. \end{equation}

Here, $\phi _z(k_2,\omega )$ is the spanwise wavenumber–frequency spectrum, while $\phi _x(k_1,\omega )$ denotes the effects of both the coherence decay and the convection of turbulent eddies. Thus $S_{pp}$ can be shown to be

(4.6) \begin{align}& S_{pp}(\boldsymbol{X}_0,\omega)\notag\\ &\quad =\left(\dfrac{\omega Y_0c}{4{\rm \pi} c_0S_0^2}\right)^22{\rm \pi} d\sum_{m=-\infty}^{\infty}\int_{-\infty}^{\infty}|\mathcal{L}(k_1,2m{\rm \pi}/\lambda,\omega)|^2\,\phi_x(k_1,\omega)\,\mathrm{d}k_1\,\phi_z(2m{\rm \pi}/\lambda,\omega). \end{align}

Equation (4.6) shows that the form of $\phi _x$ plays an important role in estimating the integral and hence the far-field sound.

Under the frozen turbulence assumption, as discussed in § 2, the streamwise coherence function is equal to 1, and the convection velocity is assumed to be a constant value. This implies that all eddies convect at the same velocity while maintaining their coherence. As a result, the cross-spectral density is found to be

(4.7)\begin{equation} G_{pp}(\xi,\eta,\omega)=G_{pp}(0,\eta,\omega)\,\mathrm{e}^{-\mathrm{i}\omega\xi/U_c}. \end{equation}

It follows that

(4.8)\begin{equation} \phi_x(k_1,\omega)=\delta(k_1-\omega/U_c). \end{equation}

Substituting (4.8) into (4.6), we recover

(4.9)\begin{equation} S_{pp}(\boldsymbol{X}_0,\omega)=\left(\dfrac{\omega Y_0c}{4{\rm \pi} c_0S_0^2}\right)^22{\rm \pi} d\sum_{m=-\infty}^{\infty}|\mathcal{L}(\bar{k}_1,2m{\rm \pi}/\lambda,\omega)|^2\,\phi_z(2m{\rm \pi}/\lambda,\omega), \end{equation}

where $\bar {k}_1=\omega /U_c$. Equation (4.9) is identical to (2.56) in Lyu et al. (Reference Lyu, Azarpeyvand and Sinayoko2016). In this case, a given frequency $\omega$ is assumed to be associated with a specific wavenumber $\bar {k}_1$ through the convection velocity $U_c$. Therefore, only the convection of eddies is considered, while the streamwise distortion is neglected. In terms of noise prediction models, the sound response function $\mathcal {L}$ sees only the value at the convective wavenumber $\bar {k}_1$. This simplification may be the reason why most analytical models overestimate the noise reduction of serrated trailing edges. It is worth noting that in the present work, the frozen turbulence refers to the idea that the wall pressure fluctuation pattern convects downstream as if it is frozen, which is widely used in the TE noise modelling. Since the wall pressure is the imprint of turbulence of various scales within the boundary layer, the frozen pressure fluctuation pattern implies that eddies of different sizes must convect at the same speed. Otherwise, the pressure fluctuation pattern would change and a coherence decay would occur.

To account for the non-frozen nature of the turbulent boundary layer quantitatively, a coherence decay function is needed. We use

(4.10)\begin{equation} \gamma(\xi,0,\omega)=\mathrm{e}^{-|\xi|/l_x(\omega)}, \end{equation}

as informed by (3.12). By employing this approximation, we can evaluate the integral in (4.6) and obtain

(4.11)\begin{equation} \phi_x(k_1,\omega)=\dfrac{l_x(\omega)}{{\rm \pi}[1+(k_1-\omega/U_c(\omega))^2\,l_x^2(\omega)]}. \end{equation}

Equation (4.11) incorporates the decay of streamwise coherence caused by the distortion of turbulent eddies as they convect downstream. It can be seen that wavenumbers around $\omega /U_c(\omega )$ still play a significant role in shaping the spectrum; however, the contribution to the spectrum is not limited to a one-to-one correspondence between a given frequency and a specific wavenumber. This spreading phenomenon is a notable spectral characteristic of the wall pressure fluctuations beneath a turbulent boundary. In the following analysis, we adopt the notation $\tilde {k}_1(\omega )$ to represent the frequency-dependent convective wavenumber, i.e. $\tilde {k}_1(\omega )=\omega /U_c(\omega )$. This parameter represents the dominant wavenumber on the frequency of $\omega$.

To obtain the acoustic prediction, we need to evaluate the integral of $|\mathcal {L}|^2\phi _x$ over the streamwise wavenumber $k_1$. As mentioned at the beginning of this subsection, rather than relying on numerical techniques, we aim to obtain a compact estimation for the integral, enabling a convenient prediction and determining the physical impact of non-frozen turbulence on TE noise.

4.2. Approximation of the model

According to the mean value theorem for integrals (Stewart Reference Stewart2011), for a given frequency $\omega$ and spanwise mode $m$, there exists a characteristic streamwise wavenumber $k_{1,m}^\ast (\omega )$ such that

(4.12)\begin{align} \int_{-\infty}^{\infty}|\mathcal{L}(k_1,2m{\rm \pi}/\lambda,\omega)|^2\,\phi_x(k_1,\omega) \,\mathrm{d}k_1=|\mathcal{L}(k_{1,m}^\ast(\omega),2m{\rm \pi}/\lambda,\omega)|^2 \int_{-\infty}^{\infty}\phi_x(k_1,\omega)\,\mathrm{d}k_1. \end{align}

Substituting the expression for $\phi _x$ given in (4.11) into (4.12) and recognizing that

(4.13)\begin{equation} \int_{-\infty}^{\infty}\dfrac{l_x(\omega)}{{\rm \pi}[1+(k_1-\tilde{k}_1(\omega))^2\,l_x^2(\omega)]}\,\mathrm{d}k_1=1, \end{equation}

we have

(4.14)\begin{equation} \int_{-\infty}^{\infty}|\mathcal{L}(k_1,2m{\rm \pi}/\lambda,\omega)|^2\,\phi_x(k_1,\omega)\,\mathrm{d}k_1 =|\mathcal{L}(C_m(\omega)\,\tilde{k}_1(\omega),2m{\rm \pi}/\lambda,\omega)|^2. \end{equation}

Substituting (4.14) into (4.6), we see that the resulting non-frozen model is identical to the frozen model apart from the introduction of a correction coefficient $C_m(\omega )=k_{1,m}^\ast (\omega )/\tilde {k}_1(\omega )$ to the dominant convection wavenumber. For a given frequency, by determining the correction coefficient $C_m$, the far-field noise can be calculated using essentially the same frozen prediction model. However, due to the complex nature of the response function, determining this coefficient is challenging. Therefore, in subsequent analysis, we aim to find a convenient approximation for $|\mathcal {L}|^2$. From the analysis of Lyu et al. (Reference Lyu, Azarpeyvand and Sinayoko2016), it is known that

(4.15)\begin{equation} |\mathcal{L}|^2\sim O(|a_m|^2), \end{equation}

where

(4.16)\begin{equation} a_m=\dfrac{\mathrm{e}^{\mathrm{i}m{\rm \pi}/2}}{2}\,\mathrm{sinc}(k_1h-m{\rm \pi}/2) +\dfrac{\mathrm{e}^{-\mathrm{i}m{\rm \pi}/2}}{2}\,\mathrm{sinc}(k_1h+m{\rm \pi}/2). \end{equation}

Examining the shape of $|a_m|^2$, we see that it can be very well approximated by

(4.17)\begin{equation} H_m(k_1h)=\dfrac{1}{4}\left(\dfrac{1}{(k_1h+m{\rm \pi}/2)^2+\tilde{m}} +\dfrac{1}{(k_1h-m{\rm \pi}/2)^2+\tilde{m}}\right), \end{equation}

where

(4.18)\begin{equation} \tilde{m}=1-\dfrac{1}{m{\rm \pi}+2}. \end{equation}

It can be seen that $H_m$ is a purely algebraic function of $k_1h$. Therefore, the correction coefficient can be determined analytically by solving

(4.19)\begin{equation} H_m(C_m\tilde{k}_1h)=\int_{-\infty}^{\infty}H_m(k_1h)\,\phi_x(k_1,\omega)\,\mathrm{d}k_1. \end{equation}

The integral in (4.19) can be found analytically to be $I_m/4$, and the expression for $I_m$ can be written as

(4.20a)$$\begin{gather} I_m=\dfrac{I_{m1}+I_{m2}}{I_{m3}+I_{m4}}+\dfrac{I_{m5}+I_{m6}}{I_{m7}+I_{m8}}, \end{gather}$$

(4.20b)$$\begin{gather}I_{m1}=4\sqrt{\tilde{m}}(4\tilde{m}+{\rm \pi}^2 m^2+4{\rm \pi} m\sigma_2+4\sigma_2^2-4\sigma_1^2), \end{gather}$$

(4.20c)$$\begin{gather}I_{m2}=4(-4\tilde{m} \sigma_1+{\rm \pi}^2 m^2\sigma_1+4{\rm \pi} m\sigma_1\sigma_2+4\sigma_1\sigma_2^2+4\sigma_1^3), \end{gather}$$

(4.20d)$$\begin{gather}I_{m3}=\sqrt{\tilde{m}}\,(16\tilde{m}^2+8({\rm \pi} m+2\sigma_2-2\sigma_1 )({\rm \pi} m+2\sigma_2+2\sigma_1)\tilde{m}+{\rm \pi}^4 m^4+8{\rm \pi}^3\sigma_2 m^3), \end{gather}$$

(4.20e)$$\begin{gather}\begin{aligned}I_{m4}&=\sqrt{\tilde{m}}\,(24{\rm \pi}^2 m^2\sigma_2^2+8{\rm \pi}^2 m^2\sigma_1^2+32{\rm \pi} m\sigma_2^3+32{\rm \pi} m\sigma_1^2\sigma_2\notag\\ &\quad +16\sigma_2^4+32\sigma_1^2\sigma_2^2+16\sigma_1^4), \end{aligned}\end{gather}$$

(4.20f)$$\begin{gather}I_{m5}=4\sqrt{\tilde{m}}\,(4\tilde{m}+{\rm \pi}^2 m^2-4{\rm \pi} m\sigma_2+4\sigma_2^2-4\sigma_1^2), \end{gather}$$

(4.20g)$$\begin{gather}I_{m6}=4(-4\tilde{m} \sigma_1+{\rm \pi}^2 m^2\sigma_1-4{\rm \pi} m\sigma_1\sigma_2+4\sigma_1\sigma_2^2+4\sigma_1^3), \end{gather}$$

(4.20h)$$\begin{gather}I_{m7}=\sqrt{\tilde{m}}\,(16\tilde{m}^2+8({\rm \pi} m-2\sigma_2-2\sigma_1 )({\rm \pi} m-2\sigma_2+2\sigma_1)\tilde{m}+{\rm \pi}^4 m^4-8{\rm \pi}^3\sigma_2 m^3), \end{gather}$$

(4.20i)$$\begin{gather}\begin{aligned}I_{m8}&=\sqrt{\tilde{m}}\,(24{\rm \pi}^2 m^2\sigma_2^2+8{\rm \pi}^2 m^2\sigma_1^2-32{\rm \pi} m\sigma_2^3-32{\rm \pi} m\sigma_1^2\sigma_2\notag\\ &\quad +16\sigma_2^4+32\sigma_1^2\sigma_2^2+16\sigma_1^4), \end{aligned} \end{gather}$$

where $\sigma _1=h/l_x$ and $\sigma _2=\tilde {k}_1h$. Subsequently, the correction coefficient $C_m$ can be computed as

(4.21)\begin{equation} C_m=\left.\sqrt{\dfrac{m^2{\rm \pi}^2}{4}-\tilde{m}+\dfrac{1+\sqrt{1+m^2{\rm \pi}^2I_m-m^2{\rm \pi}^2\tilde{m} I_m^2}}{I_m}}\right/\sigma_2. \end{equation}

It can be seen that the correction coefficient $C_m$ is determined by $\sigma _1$ and $\sigma _2$ only. Note, however, that introducing a correction coefficient is simply to facilitate a quick evaluation. Therefore, we do not expect $C_m$ to be strictly accurate.

With the introduction and evaluation of $C_m$, the far-field noise spectrum is shown to be

(4.22)\begin{align} S_{pp}(\boldsymbol{X}_0,\omega)=\left(\dfrac{\omega Y_0c}{4{\rm \pi} c_0S_0^2}\right)^22{\rm \pi} d\sum_{m=-\infty}^{\infty}|\mathcal{L}(C_m(\omega)\,\tilde{k}_1(\omega),2m{\rm \pi}/\lambda,\omega)|^2\,\phi_z(2m{\rm \pi}/\lambda,\omega). \end{align}

Equation (4.22) is purposely cast into the same form as the frozen model so that the effects of non-frozen turbulence can be accounted for conveniently by a single correction coefficient $C_m(\omega )$. In the following subsections, we will show the prediction results obtained using (4.22), along with a discussion of the rationality behind the approximations used in this subsection.

4.3. Prediction results

Using the new model that incorporates the impact of non-frozen turbulence, we can now predict the far-field noise. We apply the model to flat plates with straight and serrated trailing edges, and use parameters similar to those employed in the preceding numerical simulations. In particular, the Mach number is chosen to be $M_0=0.03$, while the chord length of the flat plate is chosen as $c=1.12$ m. The streamwise correlation length $l_x(\omega )$ and the spanwise wavenumber–frequency spectrum $\phi _z(k_2,\omega )$ are both obtained from the numerical simulation. Smol'yakov's model is selected to compute the frequency-dependent convection velocity for the new model. For the frozen model, the same computed spanwise wavenumber–frequency spectrum and a constant convection velocity $U_c=0.7U_0$ are used. The non-dimensionalized form of the far-field power spectral density,

(4.23)\begin{equation} \varPsi(\boldsymbol{X},\omega)=\dfrac{2{\rm \pi}\, S_{pp}(\boldsymbol{X},\omega)}{C_\ast(\rho_0v_\ast^2)^2\,(d/c_0)}, \end{equation}

is used to facilitate a direct comparison with results from frozen models, where $C_\ast \approx 0.1553$ and $v_\ast \approx 0.03U_0$.

Figure 9 presents the predicted far-field noise using both the frozen and non-frozen models. Here, we use $kc$ to represent the dimensionless frequency, where $k=\omega /c_0$. The observer is located at $90^\circ$ above the trailing edge, and the distance between the observer and the trailing edge is equal to the chord length $c$. The serration amplitudes are set to $2h/c=0.05$ and $2h/c=0.1$, while the aspect ratio remains constant, i.e. $\lambda /h=0.1$. It can be seen that there are minimal discrepancies between the results obtained using these two models for straight trailing edges. As will be discussed in the following analysis, the correction due to non-frozen turbulence goes to zero for straight trailing edges. The slight difference between the two curves is attributed to the frequency-dependent convection velocity used in the new model. Figure 9 shows that the frequency-dependent variation of the convection velocity has a limited influence. Significant noise reductions predicted by the frozen model can be seen within the intermediate-frequency range for serrated trailing edges, especially for the longer serration. However, the new model that accounts for non-frozen turbulence predicts less pronounced noise reductions. One important feature shown in figure 9 is that the noise reduction level reduces at high frequencies, which will be discussed further in the rest of the paper. It is also shown that the long serration with the same slope performs better than the short one.

Figure 9. Predicted far-field noise produced by the straight and serrated trailing edges: (a) $\lambda /h=0.1$, $2h/c=0.05$, and (b) $\lambda /h=0.1$, $2h/c=0.1$.

The directivity patterns predicted by frozen and non-frozen models for different Mach numbers and frequencies are shown in figure 10. The frozen model assumes a constant convection velocity $U_c=0.7U_0$. It can be seen that the presence of serrated trailing edges significantly influences the directivity patterns, especially at higher frequencies (see figures 10b,d). Similar to those observed in figure 9, the new model predicts reduced levels of noise reduction, while the shapes of the directivity patterns remain similar. From figure 10, we can also see that the noise reduction effects are more pronounced in the regions located in front of and behind the flat plate.

Figure 10. Directivity patterns for straight and serrated trailing edges predicted using frozen and non-frozen models: (a) $\lambda /h=0.1$, $2h/c=0.05$, $kc=6.5$; (b) $\lambda /h=0.1$, $2h/c=0.05$, $kc=22.5$; (c) $\lambda /h=0.1$, $2h/c=0.1$, $kc=6.5$; and (d) $\lambda /h=0.1$, $2h/c=0.1$, $kc=22.5$.

In the prediction results shown above, the frequency-dependent correlation length and the spanwise wavenumber–frequency spectrum are obtained through numerical simulations. However, considering the high cost associated with computational fluid dynamics or experimental investigations, the use of empirical models may be more convenient for practical applications. For example, we can use Chase's model to estimate the spanwise wavenumber–frequency spectrum, which is given by (Chase Reference Chase1987; Howe Reference Howe1991b)

(4.24)\begin{equation} \phi_z(k_2,\omega)\approx\dfrac{4C_\ast\rho_0^2v_\ast^4(\omega/U_c)^2\delta^4}{U_c\left(((\omega/U_c)^2+k_2^2)\delta^2+\chi^2\right)^2}. \end{equation}

Here, $\chi \approx 1.33$, and the boundary layer thickness $\delta$ is estimated using $\delta /c=0.382\,Re_c^{-1/5}$, where $Re_c$ represents the Reynolds number based on the chord length. Again, the Smol'yakov model (Smol'yakov Reference Smol'yakov2006) can be used to obtain the streamwise correlation length. Therefore, the new model demonstrates applicability across a broader range of scenarios.

To further investigate the accuracy of the new model, a comparison is conducted between the predicted noise reduction and that from experimental measurements by Gruber (Reference Gruber2012). Predictions from earlier models are also included for comparison. The experimental study used a NACA 65(12)-10 aerofoil at a $5^\circ$ angle of attack with free-stream velocity $U_0=40\ {\rm m}\ {\rm s}^{-1}$. The aerofoil has chord length 0.15 m, and the serration amplitude is $2h/c=0.2$. Two different sizes of serrations were used, namely $\lambda /h=0.6$ and $\lambda /h=0.1$. The observer is located at $90^\circ$ above the aerofoil.

Figure 11 shows the comparison of the noise reduction $\Delta \mathrm {SPL}=10\log _{10}(S_{pp}|_b/S_{pp}|_s)$, where $S_{pp}|_b$ and $S_{pp}|_s$ denote the far-field spectral density for the straight and serrated trailing edges, respectively. In the experiments, noise reductions are observed at intermediate frequencies, albeit not exceeding approximately 5 dB. Conversely, noise increment appears at high frequencies. This phenomenon is consistent with the observations reported in the experiments conducted by Oerlemans et al. (Reference Oerlemans, Fisher, Maeder and Kögler2009), which investigated full-scale serrated wind turbine blades. Howe's model exhibits a significant overprediction of the noise reduction, reaching a maximum $\Delta {\rm SPL}$ of $30$ dB for the narrow serration (see figure 11b). On the other hand, Lyu's original model demonstrates a more realistic prediction, but overestimation is still pronounced. Comparatively, the discrepancies between the prediction using the new model and the experimental measurements are considerably smaller for both serrations. Therefore, we see that the frozen turbulence assumption contributes significantly to the overestimation of noise reduction. Interestingly, as shown in figure 11(a), noise increase at high frequencies is also predicted by the new model. This increment phenomenon has been attributed to the presence of a cross-flow between serration teeth (Gruber Reference Gruber2012). However, the present model suggests another possible cause of the noise increase at high frequencies at this observer angle.

Figure 11. Comparison of the noise reduction predictions from analytical models with experimental measurements by Gruber (Reference Gruber2012): (a) $\lambda /h=0.6$ and (b) $\lambda /h=0.1$.

It can be seen from figure 9 that the long serration with the same slope can lead to more noise reduction. Here, we explore further the effects of the serration geometry by considering an extensive set of the serration amplitude and wavelength. Figure 12 shows the comparison of the noise reduction as a function of the normalized half-amplitude $h/\lambda$ between the analytical models and Gruber's experiments (Gruber Reference Gruber2012). According to Gruber (Reference Gruber2012), the serration wavelength remains unchanged, and the Strouhal number is defined as $St=f\delta /U_0$. The incoming flow velocity is $40\ {\rm m}\ {\rm s}^{-1}$, while the angle of attack of the aerofoil is $0^\circ$. The noise reduction is calculated using equation (2.2) of Gruber (Reference Gruber2012). As can be seen from figure 12(a), for $St=0.17$, the measured noise reduction increases with the increase of the half-amplitude. Both Lyu's model using the frozen turbulence assumption and the new model yield similar noise reduction results compared to the experiment for large amplitudes. However, when $h/\lambda$ is small, noise increases are predicted by both models, which is possibly caused by the inaccurate Chase's model. From figure 12(b), we can see that with the increase of the Strouhal number, the predicted noise reduction agrees much better with experimental results compared to the frozen model. In general, the noise reductions predicted by the two models increase with the increase of the half-amplitude. The apparent deviation in figures 12(c,d) between the new model and the experimental result is likely due to the new flow features caused by serrations that are not included in the model, as already discussed for figure 11.

Figure 12. Comparison of noise reductions predicted by analytical models and obtained from the experiments for four values of the Strouhal number: (a) $St=0.17$, (b) $St=0.45$, (c) $St=0.9$ and (d) $St=1.4$.

Figure 13 shows the comparison of the noise reduction predicted by analytical models and measured from experiments by Gruber (Reference Gruber2012) for four aspect ratios, i.e. 0.1, 0.2, 0.47 and 0.83. The serration amplitude takes a constant value. It is shown from figure 13(a) that the noise reduction measured from experiments generally increases with the decrease of the aspect ratio. The analytical models predict similar trends, as shown in figures 13(b,c). Both the experimental and analytical results show that with the decrease of aspect ratio from 0.2 to 0.1, the increase of noise reduction is not significant in the intermediate-frequency range. The noise reduction predicted by the new model is confined to approximately 8 dB, which is consistent with the experimental measurements. However, up to 15 dB noise reduction is predicted by the frozen model, which is much larger than in experiments. Notably, the non-frozen model predicted a decrease trend at high frequencies, which can also be seen in Gruber's experiments. On the contrary, the noise reduction predicted by the frozen model continually increases in the high-frequency range. It is worth noting that since the empirical wavenumber–frequency model would inevitably deviate from the experiment, achieving a quantitative agreement between the non-frozen model and experiments within the whole frequency range is quite challenging. In addition, the prediction model does not account for the additional flow features introduced by serrations, leading to discrepancies with experimental measurements, especially at high frequencies. However, figures 11–13 clearly show that including the impact of non-frozen turbulence can significantly improve the predicted noise reductions, particularly within the frequency range of validity.

Figure 13. Comparison of noise reductions predicted by analytical models and obtained from the experiments for four values of $\lambda$: (a) Gruber's experiments; (b) predicted using the new model; and (c) predicted using Lyu's original model.