2.1 Poisson equation solution

The Poisson equation governing the pressure fluctuations in a turbulent boundary layer derives from the divergence of the incompressible momentum equation, introducing Reynolds decomposition into mean and fluctuating quantities, then subtracting the time-averaged equation. As a result, the Laplacian of pressure fluctuations is equal to the sum of two source terms,

(2.1) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D6FB}^{2}\,p=\underbrace{-\,2\frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}x_{j}}}_{\mathit{turbulence-mean~shear}}\underbrace{-\,\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{j}}(u_{i}u_{j}-\overline{u_{i}u_{j}})}_{\mathit{turbulence-turbulence}},\end{eqnarray}$$

where $i,j=1,2,3$ , with $1$ the streamwise, $2$ the wall-normal and $3$ the transverse direction. According to (2.1) the pressure fluctuation, $p$ , can be considered as generated by two separate mechanisms (see the review by Gerolymos, Sénéchal & Vallet (Reference Gerolymos, Sénéchal and Vallet2013)). One mechanism is the turbulence–mean shear interaction, expressing the fact that the corresponding pressure fluctuations react immediately to a change of the mean flow gradient and are linear with respect to velocity fluctuations. The other mechanism is the nonlinear interaction of different components of the velocity fluctuations, which is not immediately affected by a change in mean flow gradient. Equation (2.1) is solved with the Dirichlet boundary condition

(2.2) $$\begin{eqnarray}\lim _{x_{2}\rightarrow \infty }p=0,\end{eqnarray}$$

whereby the pressure fluctuation vanishes at an infinite distance from the wall and the overall pressure tends to its mean free-stream value, and the Neumann boundary condition

(2.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{2}}\bigg|_{x_{2}=0}=0\end{eqnarray}$$

following from the rigid wall assumption. The space-domain solution of (2.1) is expressed within a Green’s function approach as

(2.4) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70C}}\,p(\boldsymbol{x},t)=\int _{\boldsymbol{x}^{\prime }}-\left[2\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\left(\frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}x_{j}}\,u_{j}\right)\,G(\boldsymbol{x},\boldsymbol{x}^{\prime })+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{j}}(u_{i}u_{j}-\overline{u_{i}u_{j}})\,G(\boldsymbol{x},\boldsymbol{x}^{\prime })\right]\,\text{d}\boldsymbol{x}^{\prime },\end{eqnarray}$$

where the function

(2.5) $$\begin{eqnarray}G(\boldsymbol{x},\boldsymbol{x}^{\prime })=-\frac{1}{2\,\unicode[STIX]{x03C0}}\frac{1}{\Vert \boldsymbol{x}-\boldsymbol{x}^{\prime }\Vert }\end{eqnarray}$$

satisfies the boundary conditions for an impulse source. The partial derivatives applied to the velocity variables in (2.4) can be transferred to the Green’s function by repeated applications of integration by parts and of the divergence theorem, yielding the equivalent formulations

(2.6) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70C}}\,p(\boldsymbol{x},t)=\int _{\boldsymbol{x}^{\prime }}-\left[2\left(\frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}x_{j}}\,u_{j}\right)\,\frac{\unicode[STIX]{x2202}G(\boldsymbol{x},\boldsymbol{x}^{\prime })}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}(u_{i}u_{j}-\overline{u_{i}u_{j}})\,\frac{\unicode[STIX]{x2202}G(\boldsymbol{x},\boldsymbol{x}^{\prime })}{\unicode[STIX]{x2202}x_{j}}\right]\,\text{d}\boldsymbol{x}^{\prime }\end{eqnarray}$$

and

(2.7) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70C}}\,p(\boldsymbol{x},t)=\int _{\boldsymbol{x}^{\prime }}-\left[2\left(\frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}x_{j}}\,u_{j}\right)\,\frac{\unicode[STIX]{x2202}G(\boldsymbol{x},\boldsymbol{x}^{\prime })}{\unicode[STIX]{x2202}x_{i}}+(u_{i}u_{j}-\overline{u_{i}u_{j}})\,\frac{\unicode[STIX]{x2202}^{2}G(\boldsymbol{x},\boldsymbol{x}^{\prime })}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{j}}\right]\,\text{d}\boldsymbol{x}^{\prime },\end{eqnarray}$$

where

(2.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}G(\boldsymbol{x},\boldsymbol{x}^{\prime })}{\unicode[STIX]{x2202}x_{i}}=\frac{1}{2\unicode[STIX]{x03C0}}\frac{\Vert x_{i}-x_{i}^{\prime }\Vert }{\Vert \boldsymbol{x}-\boldsymbol{x}^{\prime }\Vert ^{3}}\end{eqnarray}$$

and

(2.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}G(\boldsymbol{x},\boldsymbol{x}^{\prime })}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{j}}=\frac{1}{2\unicode[STIX]{x03C0}}\left[\frac{\unicode[STIX]{x1D6FF}_{ij}}{\Vert \boldsymbol{x}-\boldsymbol{x}^{\prime }\Vert ^{3}}-3\frac{\Vert (x_{i}-x_{i}^{\prime })(x_{j}-x_{j}^{\prime })\Vert }{\Vert \boldsymbol{x}-\boldsymbol{x}^{\prime }\Vert ^{5}}\right].\end{eqnarray}$$

Equations (2.4), (2.6) and (2.7) are mathematically equivalent, so the choice is motivated by considerations of numerical stability of the integration algorithm. Peltier & Hambric (Reference Peltier and Hambric2007) and Slama et al. (Reference Slama, Leblond and Sagaut2018) selected equation (2.7). However, Hu, Reiche & Ewert (Reference Hu, Reiche and Ewert2017) argued that (2.6) provides the best numerical realisation of the turbulence–turbulence interaction contribution. In any case, the space-domain approach prescribes a three-dimensional integral for the computation of $p$ , whereby the cross-correlation of wall-pressure fluctuations corresponds to a six-dimensional integral, which must still be Fourier transformed numerically to obtain the wavenumber–frequency spectrum that is needed in acoustic predictions. Performing these numerical integrations with limited time and computational resources would inevitably lead to a loss of accuracy. For this reason, it is preferable to derive an analytical solution for the unsteady pressure directly in the wavenumber domain.

2.2 Modified Helmholtz equation

2.2.1 Turbulence–mean shear interaction term

In this section, the wavenumber-domain expression of the unsteady pressure at the wall is derived assuming that the only significant contribution is given by the turbulence–mean shear interaction component. This approach has its roots in the analytical considerations of Kraichnan (Reference Kraichnan1956a ) and Hodgson (Reference Hodgson1961). More recently, however, Slama et al. (Reference Slama, Leblond and Sagaut2018) argued that the two components have the same order of magnitude, based on the numerical integration of (2.7). The derivation of a wavenumber solution for the turbulence–turbulence interaction term is more cumbersome and requires a set of more restrictive hypotheses in order to get a simple closed-form formulation. For this reason, the turbulence–turbulence interaction component will be presented separately in § 2.4.

Assuming homogeneous turbulence in planes parallel to the wall, equation (2.1) can be Fourier transformed in space in the $(x_{1},x_{3})$ directions, yielding the following modified Helmholtz equation:

(2.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\hat{p}(\boldsymbol{k},x_{2})}{\unicode[STIX]{x2202}x_{2}^{2}}-k^{2}\,\hat{p}(\boldsymbol{k},x_{2})=-2\,\unicode[STIX]{x1D70C}\,\text{i}\,k_{i}\,\frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}x_{j}}\hat{u} _{j}(\boldsymbol{k},x_{2}),\end{eqnarray}$$

where the following convention for the Fourier transform is adopted:

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{p}(\boldsymbol{k},x_{2}) & \displaystyle =\frac{1}{4\unicode[STIX]{x03C0}^{2}}\iint _{-\infty }^{+\infty }p(\boldsymbol{x})\,\text{e}^{-\text{i}(k_{1}x_{1}+k_{3}x_{3})}\,\text{d}x_{1}\,\text{d}x_{3},\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{Q}(\boldsymbol{k},x_{2}) & \displaystyle =\frac{1}{4\unicode[STIX]{x03C0}^{2}}\iint _{-\infty }^{+\infty }Q(\boldsymbol{x})\,\text{e}^{-\text{i}(k_{1}x_{1}+k_{3}x_{3})}\,\text{d}x_{1}\,\text{d}x_{3}.\end{eqnarray}$$

Furthermore, assuming that the only non-vanishing mean velocity gradient is that of the streamwise component, $U_{1}$ , in the wall-normal direction, $x_{2}$ , the source term is reduced to

(2.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\hat{p}(\boldsymbol{k},x_{2})}{\unicode[STIX]{x2202}x_{2}^{2}}-k^{2}\,\hat{p}(\boldsymbol{k},x_{2})=-2\,\unicode[STIX]{x1D70C}\,\text{i}\,k_{1}\,\frac{\unicode[STIX]{x2202}U_{1}}{\unicode[STIX]{x2202}x_{2}}\hat{u} _{2}(\boldsymbol{k},x_{2}).\end{eqnarray}$$

It is demonstrated in appendix A that the wavenumber-domain solution for the pressure at the wall is

(2.14) $$\begin{eqnarray}\hat{p}(\boldsymbol{k},x_{2}=0)=2\,\text{i}\,\unicode[STIX]{x1D70C}\frac{k_{1}}{k}\int _{0}^{\infty }\frac{\unicode[STIX]{x2202}U_{1}(X_{2})}{\unicode[STIX]{x2202}x_{2}}\hat{u} _{2}(\boldsymbol{k},X_{2})\text{e}^{-k\,X_{2}}\,\text{d}X_{2}.\end{eqnarray}$$

The PSD of wall-pressure fluctuations is equal to the ensemble average of the product of (2.14) by its complex conjugate,

(2.15) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{pp}^{TM}(\boldsymbol{k}) & = & \displaystyle 4\unicode[STIX]{x1D70C}_{0}^{2}\int _{\boldsymbol{k}^{\prime }}\iint _{0}^{\infty }\frac{k_{1}^{2}}{k^{2}}\text{e}^{-(X_{2}+X_{2}^{\prime })k}\frac{\unicode[STIX]{x2202}U_{1}(X_{2})}{\unicode[STIX]{x2202}x_{2}}\frac{\unicode[STIX]{x2202}U_{1}(X_{2}^{\prime })}{\unicode[STIX]{x2202}x_{2}}\nonumber\\ \displaystyle & & \displaystyle \times \langle \hat{u} _{2}(\boldsymbol{k},X_{2})\hat{u} _{2}^{\ast }(\boldsymbol{k}^{\prime },X_{2}^{\prime })\rangle \text{d}X_{2}\,\text{d}X_{2}^{\prime }\,\text{d}\boldsymbol{k}^{\prime }.\end{eqnarray}$$

The ensemble average of vertical velocity fluctuations appearing in (2.15) can be expressed as

(2.16) $$\begin{eqnarray}\langle \hat{u} _{2}(\boldsymbol{k},X_{2})\hat{u} _{2}^{\ast }(\boldsymbol{k}^{\prime },X_{2}^{\prime })\rangle =\unicode[STIX]{x1D711}_{22}(\boldsymbol{k},X_{2},X_{2}^{\prime })\,\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{k}^{\prime })\end{eqnarray}$$

by virtue of the statistical orthogonality of turbulence wavevectors (see Bailly & Comte-Bellot Reference Bailly and Comte-Bellot2015). Finally, substituting equation (2.16) in (2.15) and integrating over $\text{d}\boldsymbol{k}^{\prime }$ yields

(2.17) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{pp}^{TM}(\boldsymbol{k})=4\unicode[STIX]{x1D70C}_{0}^{2}\iint _{0}^{\infty }\frac{k_{1}^{2}}{k^{2}}\text{e}^{-(X_{2}+X_{2}^{\prime })k}\frac{\unicode[STIX]{x2202}U_{1}(X_{2})}{\unicode[STIX]{x2202}x_{2}}\frac{\unicode[STIX]{x2202}U_{1}(X_{2}^{\prime })}{\unicode[STIX]{x2202}x_{2}}\unicode[STIX]{x1D711}_{22}\,\text{d}X_{2}\,\text{d}X_{2}^{\prime }.\end{eqnarray}$$

The main point in (2.17) is the correct analytical representation of the vertical velocity fluctuation cross-spectral density, $\unicode[STIX]{x1D711}_{22}$ . The remaining terms, namely the mean velocity gradient and the vertical velocity root-mean-square (r.m.s.), are relatively easier to calculate. They can be obtained from empirical models, as in Panton & Linebarger (Reference Panton and Linebarger1974), from hot-wire measurements or from steady-state simulations, as in Remmler et al. (Reference Remmler, Christophe, Anthoine and Moreau2010). Although the time variable has been omitted in the derivation of the solution of the modified Helmholtz equation, it is understood that the variable $\hat{p}$ is dependent on time. The wavenumber spectrum of (2.17) can be converted to a frequency spectrum first by integrating over the $k_{3}$ wavenumber and then by adopting Taylor’s hypothesis (see Taylor Reference Taylor1938) of frozen convection of the turbulent eddies in the flow direction. Assuming constant convection speed of wall-pressure fluctuations, $U_{c}$ , in a close range around the point where the frequency spectrum is sought, the sole streamwise wavenumber contributing to a given radian frequency, $\unicode[STIX]{x1D714}$ , is $K_{c}=\unicode[STIX]{x1D714}/U_{c}$ . Therefore, the frequency spectrum of wall-pressure fluctuations is

(2.18) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{pp}(\unicode[STIX]{x1D714})=\frac{\displaystyle \int _{-\infty }^{+\infty }\unicode[STIX]{x1D711}_{pp}^{TM}(K_{c},k_{3})\,\text{d}k_{3}}{U_{c}}.\end{eqnarray}$$

2.3 Turbulence modelling

The analytical expression of the cross-spectral density of vertical velocity fluctuations, which closes equation (2.17), is based on a theory of isotropic turbulence which, following Panton & Linebarger (Reference Panton and Linebarger1974), is modified in two ways. In the first place, the integral length scale, $\unicode[STIX]{x1D6EC}$ , is allowed to vary according to the distance from the wall. Second, the stretching of the turbulent structures in the flow direction is taken into account with a coefficient $\unicode[STIX]{x1D6FC}$ representing the ratio of streamwise to transverse integral length scales (see also Schlinker & Amiet Reference Schlinker and Amiet1981). The cross-spectral density of anisotropic turbulence, $\unicode[STIX]{x1D711}_{22}^{NI}$ , is related to the isotropic spectrum, $\unicode[STIX]{x1D711}_{22}^{I}$ , as

(2.19) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{22}^{NI}(k_{1},k_{3},x_{2},x_{2}^{\prime })=\unicode[STIX]{x1D6FC}\,\unicode[STIX]{x1D711}_{22}^{I}(\unicode[STIX]{x1D6FC}\,k_{1},k_{3},x_{2},x_{2}^{\prime }).\end{eqnarray}$$

The ratio of length scales, $\unicode[STIX]{x1D6FC}$ , varies across the boundary layer with the distance from the wall, $x_{2}$ . Remmler et al. (Reference Remmler, Christophe, Anthoine and Moreau2010) postulate the existence of a function relating each wavenumber $k_{1}$ to a given value of $\unicode[STIX]{x1D6FC}$ . However, the wall-pressure PSD at a given wavenumber $k_{1}$ results from a double integration over the distance from the wall (equation (2.17)) so that the result at each $k_{1}$ does not depend on a single value of $\unicode[STIX]{x1D6FC}$ , but rather on the variation of $\unicode[STIX]{x1D6FC}$ with $x_{2}$ .

A characteristic of boundary-layer turbulence that is not included in this model is the $45^{\circ }$ orientation of the vorticity vectors due to mean straining, the effect of which was deemed by Kraichnan (Reference Kraichnan1956b ) negligible with respect to that of length scale anisotropy. However, this aspect has been taken into account in the models of Peltier & Hambric (Reference Peltier and Hambric2007) and Slama et al. (Reference Slama, Leblond and Sagaut2018).

The expression of $\unicode[STIX]{x1D711}_{22}^{I}$ follows from the choice of the analytical expression of the longitudinal correlation function of turbulence, $F(r)$ (see (B 13) and (B 17)). Two analytical expressions are considered in this work: the Gaussian and the generalised von Kármán function described in the following. The complete derivation of the corresponding cross-spectral density expressions can be found in appendix B.

2.3.1 Gaussian energy spectrum

The Gaussian longitudinal correlation function is defined as

(2.20) $$\begin{eqnarray}F(r)=\exp \left(-\frac{r^{2}}{L^{2}}\right),\end{eqnarray}$$

with the characteristic length

(2.21) $$\begin{eqnarray}L=\frac{2}{\sqrt{\unicode[STIX]{x03C0}}}\unicode[STIX]{x1D6EC},\end{eqnarray}$$

complying with the definition of $\unicode[STIX]{x1D6EC}$ given in (B 3). The corresponding cross-spectral density of vertical velocity fluctuations is

(2.22) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{22}(\boldsymbol{k},x_{2},x_{2}^{\prime })=\frac{\sqrt{\overline{u_{2}^{2}}(x_{2})\,\overline{u_{2}^{2}}(x_{2}^{\prime })}\,L^{4}}{16\unicode[STIX]{x03C0}}\,k^{2}\,\exp \left(-\frac{L^{2}\,k^{2}}{4}-\frac{(x_{2}-x_{2}^{\prime })^{2}}{L^{2}}\right).\end{eqnarray}$$

This is possibly the simplest mathematical formulation available in the literature. However, its accuracy has been questioned already by Panton & Linebarger (Reference Panton and Linebarger1974) and, more recently, by Kamruzzaman et al. (Reference Kamruzzaman, Lutz, Würz and Krämer2011), for instance.

2.3.2 Generalised von Kármán energy spectrum

The generalised von Kármán correlation function is defined by Wilson (Reference Wilson1997) as

(2.23) $$\begin{eqnarray}F(r)=\frac{1}{2^{\unicode[STIX]{x1D708}-1}\,\unicode[STIX]{x0393}(\unicode[STIX]{x1D708})}\left(\frac{r}{l}\right)^{\unicode[STIX]{x1D708}}\text{K}_{\unicode[STIX]{x1D708}}\left(\frac{r}{l}\right),\end{eqnarray}$$

with the characteristic length

(2.24) $$\begin{eqnarray}l=\frac{\unicode[STIX]{x0393}(\unicode[STIX]{x1D708})}{\sqrt{\unicode[STIX]{x03C0}}\,\unicode[STIX]{x0393}(\unicode[STIX]{x1D708}+1/2)}\unicode[STIX]{x1D6EC}.\end{eqnarray}$$

The corresponding cross-spectral density of vertical velocity fluctuations is

(2.25) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{22}\left(\boldsymbol{k},x_{2},x_{2}^{\prime }\right)=\frac{\sqrt{\overline{u_{2}^{2}}(x_{2})\,\overline{u_{2}^{2}}(x_{2}^{\prime })}\,l^{2}\,\tilde{k}^{2}\,\unicode[STIX]{x1D701}^{\unicode[STIX]{x1D708}+2}}{\unicode[STIX]{x0393}(\unicode[STIX]{x1D708})\,\unicode[STIX]{x03C0}\,2^{\unicode[STIX]{x1D708}+1}\,\left(1+\tilde{k}^{2}\right)^{\unicode[STIX]{x1D708}+2}}\text{K}_{\unicode[STIX]{x1D708}+2}(\unicode[STIX]{x1D701}),\end{eqnarray}$$

with the parameter

(2.26) $$\begin{eqnarray}\unicode[STIX]{x1D701}=\frac{\Vert x_{2}-x_{2}^{\prime }\Vert }{l}\sqrt{1+l^{2}k^{2}}.\end{eqnarray}$$

The limit of (2.25) for $x_{2}\rightarrow x_{2}^{\prime }$ is the generalised auto-spectrum

(2.27) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{22}^{a}(\boldsymbol{k},x_{2})=\frac{\unicode[STIX]{x1D708}(\unicode[STIX]{x1D708}+1)\,\overline{u_{2}^{2}}(x_{2})\,l^{4}\,k^{2}}{\unicode[STIX]{x03C0}(1+l^{2}k^{2})^{\unicode[STIX]{x1D708}+2}}.\end{eqnarray}$$

This formulation is particularly interesting for its flexibility: different values of the parameter $\unicode[STIX]{x1D708}$ represent some of the most widely used turbulence models. First, $\unicode[STIX]{x1D708}=1/3$ corresponds to the original model of von Kármán (Reference von Kármán1948). This turbulence model is also at the core of the TNO-Blake family of models, as will be discussed in the next section.

It can be easily verified that, for $\unicode[STIX]{x1D708}=1/2$ , equation (2.23) corresponds to

(2.28) $$\begin{eqnarray}F(r)=\exp \left(-\frac{r}{\unicode[STIX]{x1D6EC}}\right).\end{eqnarray}$$

This exponential longitudinal correlation function is the basis of Liepmann’s turbulence model (see Liepmann, Laufer & Liepmann Reference Liepmann, Laufer and Liepmann1951). Interestingly, also Panton & Linebarger (Reference Panton and Linebarger1974) develop their formulation of the cross-spectral density of vertical velocity fluctuations from (2.28). However, instead of simply using (2.25) with $\unicode[STIX]{x1D708}=1/2$ , they chose to formulate $\unicode[STIX]{x1D711}_{22}$ as a double integral (see (B 10)). As a result, equation (2.18) becomes a five-dimensional integral, which can be more challenging from a numerical point of view. A possible method for the fast and accurate integration of this kind of high-dimensional formulation is presented in § 5.

Finally, for $\unicode[STIX]{x1D708}=7/6$ the generalised formulation corresponds to the rapid distortion theory (RDT) proposed by Hunt (Reference Hunt1973). Although this model was originally developed to account for the modification of the energy spectrum when homogeneous turbulent structures are distorted and elongated by impingement on a bluff body, it has recently been applied by Goldstein, Leib & Afsar (Reference Goldstein, Leib and Afsar2017) to the study of the noise scattered by the trailing edge of a flat plate interacting with a jet parallel to its surface. Furthermore, the hypothesis that the turbulence–turbulence interaction is negligible with respect to turbulence–mean shear interaction is at the basis of RDT as well (see Bailly & Comte-Bellot Reference Bailly and Comte-Bellot2015). It has been suggested by Christophe (Reference Christophe2011) that the RDT formulation may be more accurate than the classical von Kármán spectrum in the low-wavenumber range (see also De Santana et al. Reference Santana, Christophe, Schram and Desmet2016). In his dissertation de la Riva (Reference de la Riva2001) has also shown the capabilities of RDT in predicting the development of turbulence convected through a highly staggered cascade propulsor configuration formed by non-symmetric aerofoils (see also de la Riva et al. Reference de la Riva, Devenport, Muthanna and Glegg2004).

The longitudinal correlation functions presented in this section are depicted in figure 1. The correlation curves tend to drop faster to zero going from the Gaussian to the von Kármán function. Furthermore, according to (B 3), the integral of every curve for $r$ going from zero to infinity is equal to $\unicode[STIX]{x1D6EC}$ , so that the curves that go faster to zero show higher correlation values for small $r$ .

Figure 1.

Analytical longitudinal correlation functions.

2.3.3 TNO-Blake model

The family of TNO-Blake models makes use of the von Kármán turbulence spectrum in order to compute the wall-pressure spectrum. However, the original TNO-Blake model (see Parchen Reference Parchen1998; Bertagnolio et al. Reference Bertagnolio, Fischer and Zhu2014) approximates the cross-spectrum of vertical velocity fluctuations as

(2.29) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{22}(k_{1},x_{2},x_{2}^{\prime },k_{3})\approx L_{2}(x_{2})\,\unicode[STIX]{x1D711}_{22}^{a}(k_{1},x_{2},k_{3})\,\unicode[STIX]{x1D6FF}(x_{2}-x_{2}^{\prime })\,\unicode[STIX]{x1D6F7}_{m}(\unicode[STIX]{x1D714}-U_{c}(x_{2})k_{1}),\end{eqnarray}$$

where $L_{2}(x_{2})$ is a vertical correlation length scale, and $\unicode[STIX]{x1D6F7}_{m}$ is the so-called moving-axis spectrum, representing the contribution of the wavenumber $k_{1}$ to a given frequency $\unicode[STIX]{x1D714}$ . In most cases, Taylor’s hypothesis is applied and the moving-axis spectrum is expressed as

(2.30) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{m}(\unicode[STIX]{x1D714}-U_{c}(x_{2})k_{1})=\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}-U_{c}(x_{2})k_{1}).\end{eqnarray}$$

The fundamental assumption behind the development of (2.29) is that

(2.31) $$\begin{eqnarray}R_{22}(r_{1},x_{2},x_{2}^{\prime },r_{3})=0\quad \text{for }x_{2}\neq x_{2}^{\prime },\end{eqnarray}$$

which can be hardly justified from a physical point of view. In fact, as will be shown later in § 4.3, the vertical velocity cross-correlation coefficient is not negligible over a vertical distance corresponding to at least two integral length scales. For this reason, Fischer et al. (Reference Fischer, Bertagnolio and Madsen2017) presented an improved version of the TNO-Blake model which corresponds to the substitution of the exact equation (2.25) (with $\unicode[STIX]{x1D708}=1/3$ ) in (2.17). However, this hypothesis has the only advantage of reducing the number of dimensions of integration of the wall-pressure spectrum formulation. Then, the substitution of (2.29) and (2.30) in (2.17) yields the wavenumber wall-pressure spectrum,

(2.32) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{pp}(\unicode[STIX]{x1D714}/U_{c},k_{3})\approx 8\,\unicode[STIX]{x1D70C}_{0}^{2}\int _{0}^{\infty }\frac{k_{1}^{2}}{k_{1}^{2}+k_{3}^{2}}\text{e}^{-2\sqrt{k_{1}^{2}+k_{3}^{2}}\,X_{2}}\left(\frac{\unicode[STIX]{x2202}U_{1}(x_{2})}{\unicode[STIX]{x2202}X_{2}}\right)^{2}L_{2}(X_{2})\,\unicode[STIX]{x1D711}_{22}^{a}(k_{1},X_{2},k_{3})\,\text{d}X_{2}.\end{eqnarray}$$

Finally, assuming equation (2.30), the corresponding frequency spectrum is calculated by coupling equation (2.32) with (2.18).

The treatment of the anisotropy of turbulence length scales in the TNO-Blake family of models is similar to that presented in (2.19). However, in this case there are three coefficients, representing the anisotropy of the length scales in the streamwise, wall-normal and transverse directions, respectively. The anisotropic three-dimensional auto-spectrum of vertical velocity fluctuations is

(2.33) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{22}^{a,NI}(k_{1},k_{2},k_{3})=\unicode[STIX]{x1D6FD}_{1}\,\unicode[STIX]{x1D6FD}_{2}\,\unicode[STIX]{x1D6FD}_{3}\,\unicode[STIX]{x1D711}_{22}^{a,I}(\unicode[STIX]{x1D6FD}_{1}k_{1},\unicode[STIX]{x1D6FD}_{2}k_{2},\unicode[STIX]{x1D6FD}_{3}k_{3}).\end{eqnarray}$$

Integrating the previous equation in $\text{d}k_{2}$ yields the two-dimensional auto-spectrum used in (2.29)

(2.34) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{22}^{a,NI}(k_{1},k_{3})=\unicode[STIX]{x1D6FD}_{1}\,\unicode[STIX]{x1D6FD}_{3}\,\unicode[STIX]{x1D711}_{22}^{a,I}(\unicode[STIX]{x1D6FD}_{1}k_{1},\unicode[STIX]{x1D6FD}_{3}k_{3}).\end{eqnarray}$$

Various values for these coefficients have been proposed over time, as can be seen in table 1. The reason for the discrepancy in the values is that each set of parameters has been tuned empirically to a different experimental database. It can be noticed that Bertagnolio et al. (Reference Bertagnolio, Fischer and Zhu2014) expressed the vertical and transverse anisotropy coefficients as a function of the free-stream pressure gradient parameter

(2.35) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{P}=\frac{\unicode[STIX]{x1D6FF}}{u_{\unicode[STIX]{x1D70F}}}\left[\frac{\text{d}P/\text{d}x_{1}}{\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x1D707}}\right],\end{eqnarray}$$

where $\text{d}P/\text{d}x_{1}$ is the gradient of the mean wall static pressure with respect to the abscissa along the aerofoil suction side. However, this dependence has been dropped in the later work of Fischer et al. (Reference Fischer, Bertagnolio and Madsen2017). None of the sets of coefficients presented in table 1 takes into account the variation of the anisotropy characteristics with the distance from the wall, a fact that is not negligible as will be shown in § 4.3.

Table 1.

Values of the anisotropy parameters $\unicode[STIX]{x1D6FD}_{i}$ according to different Authors.

The original TNO-Blake model is also characterised by the fact that the only flow quantity extracted from computational fluid dynamics (CFD) data is the mean shear, whereas the other quantities are defined with analytic expressions. These expressions can be found in Kamruzzaman et al. (Reference Kamruzzaman, Lutz, Würz, Shen, Zhu, Hansen, Bertagnolio and Madsen2012) and are here summarised. The vertical component of turbulent kinetic energy is

(2.36) $$\begin{eqnarray}\overline{u_{2}^{2}}=\unicode[STIX]{x1D6FC}_{0}\,k_{T},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{0}=0.45$ on the suction side of an aerofoil and $0.35$ on the pressure side. The turbulent kinetic energy is modelled as a function of the mean shear

(2.37) $$\begin{eqnarray}k_{T}=\frac{\sqrt{\left(l_{mix}^{2}\left|{\displaystyle \frac{\unicode[STIX]{x2202}U_{1}}{\unicode[STIX]{x2202}x_{2}}}\right|{\displaystyle \frac{\unicode[STIX]{x2202}U_{1}}{\unicode[STIX]{x2202}x_{2}}}\right)^{2}}}{0.3}\end{eqnarray}$$

and

(2.38) $$\begin{eqnarray}l_{mix}=0.085\,\unicode[STIX]{x1D6FF}\,\tanh \left(\frac{0.41x_{2}}{0.085\,\unicode[STIX]{x1D6FF}}\right).\end{eqnarray}$$

In the last equation, the coefficient $0.41$ is the von Kármán constant. Finally, the integral length scale is

(2.39) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}=\frac{l_{mix}}{0.41}.\end{eqnarray}$$

Equation (2.39) is used as an approximation of the vertical velocity correlation length $L_{2}(x_{2})$ .

2.4 A model of the turbulence–turbulence interaction wall-pressure spectrum

The derivation of a wavenumber-domain solution of the unsteady pressure generated by the turbulence–turbulence interaction source term, originally proposed by Hodgson (Reference Hodgson1961), is thoroughly reviewed and amended in some points in appendix C. In this section only the main hypotheses and the final expression are briefly recalled, referring to the appendix C for further considerations. The following assumptions are made:

1. (i) joint normal probability distribution of the turbulence at any two points across the boundary layer;

2. (ii) isotropic distribution of the turbulent kinetic energy in the three spatial coordinates

   (2.40) $$\begin{eqnarray}\overline{u_{i}^{2}}(x_{2})=\overline{u^{2}}(x_{2})={\textstyle \frac{2}{3}}k_{T}(x_{2});\end{eqnarray}$$

3. (iii) cross-spectral density of velocity fluctuations having a Gaussian distribution in the wavenumber plane, $\boldsymbol{k}$ :

   (2.41) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{ij}(x_{2},x_{2}^{\prime },\boldsymbol{k})=\frac{\overline{u^{2}}(x_{2})}{16\,\unicode[STIX]{x03C0}}\,L^{4}\,(k^{2}\unicode[STIX]{x1D6FF}_{ij}-k_{i}k_{j})\,\text{e}^{-\unicode[STIX]{x1D705}^{2}L^{2}/4}\,{\mathcal{F}}_{2},\end{eqnarray}$$
   where ${\mathcal{F}}_{2}$ represents the vertical velocity correlation function: it can be assumed as Gaussian (C 32) or an exponential function (C 33); and

4. (iv) negligible contribution of turbulence cross-spectral densities such that $\unicode[STIX]{x1D711}_{i2}$ , $i\in \{1,3\}$ .

Appendix C then shows that the following expression of the turbulence–turbulence wall-pressure PSD can be derived:

(2.42) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{pp}^{TT}(\boldsymbol{k}) & = & \displaystyle \frac{\unicode[STIX]{x1D70C}_{0}^{2}\,L^{2}}{128\,\unicode[STIX]{x03C0}}\iint _{0}^{\infty }\overline{u^{2}}(X_{2})\,\overline{u^{2}}(X_{2}^{\prime })\,\text{e}^{-k(X_{2}+X_{2}^{\prime })}\,{\mathcal{F}}_{2}^{2}\,\text{d}X_{2}\,\text{d}X_{2}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \times \,k^{2}\,(28+L^{2}k^{2})\,\text{e}^{-L^{2}k^{2}/8}.\end{eqnarray}$$

Equation (2.42) can be easily integrated with any numerical technique, being a two-dimensional integral. From this point of view, it can be compared with (2.4) to (2.7). In fact, Peltier & Hambric (Reference Peltier and Hambric2007) and Slama et al. (Reference Slama, Leblond and Sagaut2018) also made the assumption of a joint normal probability distribution of turbulence in order to treat the turbulence–mean shear and turbulence–turbulence interaction terms separately. Consequently, multiplying equation (2.7) by its complex conjugate in order to get the wall-pressure correlation coefficient yields a six-dimensional integral which must still be Fourier transformed in two dimensions in order to get the wavenumber PSD. However, the solution proposed in this section is derived at the price of some restrictive assumptions and should be seen as a means to better understand the relative importance of the two wall-pressure source components.

3.1 Comparison with Goody’s model

The empirical model developed by Goody (Reference Goody2004) represents the basic physical features of a wall-pressure spectrum under a zero-pressure-gradient turbulent boundary layer. At the lower frequencies, which are dominated by larger-scale turbulent structures present in the outer part of the boundary layer, the spectrum rises as $\unicode[STIX]{x1D714}^{2}$ . The $\unicode[STIX]{x1D714}^{-0.7}$ decay corresponds to an overlap region that collapses with both inner and outer boundary-layer scaling parameters (see Goody Reference Goody2004). It has been argued that this region of the spectrum is mostly influenced by the physics of the log layer (see Bradshaw Reference Bradshaw1967). Finally, the wall-pressure spectrum decays as $\unicode[STIX]{x1D714}^{-5}$ for $\unicode[STIX]{x1D714}\rightarrow \infty$ (see also Moreau & Roger Reference Moreau and Roger2005, p. 49). These trends are represented in the equation

(3.5) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x1D711}_{pp}(\unicode[STIX]{x1D714})\,U_{0}}{\unicode[STIX]{x1D70F}_{w}^{2}\,\unicode[STIX]{x1D6FF}}}={\displaystyle \frac{3.0\,\left({\displaystyle \frac{\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D6FF}}{U_{0}}}\right)^{2}}{\left[\left({\displaystyle \frac{\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D6FF}}{U_{0}}}\right)^{0.75}+0.5\right]^{3.7}+\left[(1.1\,R_{T}^{-0.57})\,{\displaystyle \frac{\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D6FF}}{U_{0}}}\right]^{7}}},\end{eqnarray}$$

where

$$\begin{eqnarray}R_{T}=\left(\frac{u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D708}}\right)\sqrt{\frac{C_{f}}{2}}\end{eqnarray}$$

represents the ratio of the outer layer to inner layer time scales. It can be noticed that the use of inner and outer boundary-layer parameters for normalisation provides a better overlap of different experimental curves over the whole frequency range (see Keith et al. Reference Keith, Hurdis and Abraham1992; Cipolla & Keith Reference Cipolla and Keith2000; Goody Reference Goody2004).

Equation (3.5) is compared with the distributions produced by the turbulence models of § 2.3, following the assumption that the main contribution to the wall-pressure spectrum is that of the turbulence–mean shear interaction. The results are depicted in figure 2. The Gaussian and Liepmann curves are obtained by means of (3.3) and (3.4), respectively. The wavenumber spectra given by these equations are transformed into frequency spectra by integrating numerically equation (2.18). The von Kármán and RDT curves are obtained by numerical integration of (2.17) and (2.18) in the $X_{2}$ , $X_{2}^{\prime }$ and $k_{3}$ variables. The following parameters, representative of a subsonic turbulent boundary layer at $M=0.2$ and $Re_{c}=5.85\times 10^{5}$ , have been used: $R_{T}=31.7$ , $U_{0}=68.0~\text{m}~\text{s}^{-1}$ , $U_{0}/U_{c}=1.25$ , $\overline{u^{2}}=4.16~\text{m}^{2}~\text{s}^{-2}$ , $\unicode[STIX]{x1D6FF}=3.38\times 10^{-3}~\text{m}$ , $\unicode[STIX]{x1D6EC}=0.473\times 10^{-3}~\text{m}$ , $\unicode[STIX]{x1D6FE}=1/(2\unicode[STIX]{x1D6EC})$ , $\unicode[STIX]{x1D70F}_{w}=11.8~\text{Pa}$ , $\unicode[STIX]{x1D707}=1.825\times 10^{-5}~\text{kg}~(\text{m}~\text{s})^{-1}$ and $\unicode[STIX]{x1D70C}_{0}=1.204~\text{kg}~\text{m}^{-3}$ . The results show that all turbulence models considered in this study are able to represent the quadratic rise of the wall-pressure PSD at low frequencies. In particular, the Liepmann and von Kármán curves are quite close to Goody’s in that range. Concerning the high-frequency range, only the Liepmann and von Kármán curves show a decay close to $\unicode[STIX]{x1D714}^{-5}$ , whereas the Gaussian curve drops much faster. Also the RDT curve drops slightly faster than the Liepmann and von Kármán ones. The $\unicode[STIX]{x1D714}^{-5}$ slope has been predicted theoretically by Blake (Reference Blake1986), who attributed it to the dominating influence of the viscous sub-layer. This is consistent with the fact that the Gaussian turbulence spectrum represents only large-scale turbulence structures, whereas the three turbulence spectral models classified under the generalised von Kármán spectrum take into account also the energy contained in the finer-scale structures such as those present in the viscous sub-layer. It is thus shown that the mere knowledge of the wavenumber distribution of turbulent kinetic energy, with only a simplified model of the boundary-layer mean and r.m.s. velocity profiles, is sufficient to justify the low-frequency and high-frequency slopes of the wall-pressure spectrum observed experimentally. On the contrary, the $\unicode[STIX]{x1D714}^{-0.7}$ slope of the overlap region cannot be retrieved by analytical considerations alone, at least not without a realistic model of the boundary-layer velocity profiles.

3.2 Effect on the mean-square wall pressure due to turbulence–mean shear interaction

The mean-square wall pressure due to the turbulence–mean shear interaction is defined as the integral of the wall-pressure PSD over the wavenumber spectrum

(3.6) $$\begin{eqnarray}\langle p^{2}\rangle _{TM}=\int _{\boldsymbol{k}}\unicode[STIX]{x1D711}_{pp}^{TM}(\boldsymbol{k})\,\text{d}\boldsymbol{k}.\end{eqnarray}$$

This integration can be performed analytically in order to understand how a given formulation of $\unicode[STIX]{x1D711}_{22}$ influences the predicted mean-square wall pressure, which is in turn directly proportional to the overall noise emitted by the trailing edge of an aerofoil, for instance. However, analytical solutions can be obtained only at the price of a further simplification, that is imposing $\unicode[STIX]{x1D6FE}=0$ . In this case, the mean shear would be constant across the boundary layer and the mean velocity profile would be linear. This is then representative only of the physics of the laminar sublayer of a turbulent boundary layer. The application of (3.6) to the Gaussian-based wall-pressure spectrum of (3.3) yields

(3.7) $$\begin{eqnarray}\langle p^{2}\rangle _{TM}=\frac{1}{3}\frac{\unicode[STIX]{x1D70C}_{0}^{2}\unicode[STIX]{x1D70F}_{w}^{2}\overline{u^{2}}L^{2}}{\unicode[STIX]{x1D707}^{2}}.\end{eqnarray}$$

Similarly, the integration of the Liepmann-based wall-pressure spectrum of (3.4) yields

(3.8) $$\begin{eqnarray}\langle p^{2}\rangle _{TM}=\frac{2}{3}\frac{\unicode[STIX]{x1D70C}_{0}^{2}\unicode[STIX]{x1D70F}_{w}^{2}\overline{u^{2}}\unicode[STIX]{x1D6EC}^{2}}{\unicode[STIX]{x1D707}^{2}}.\end{eqnarray}$$

It is thus verified that the flattening of the longitudinal correlation function of turbulence from a Gaussian to an exponential curve (see figure 1) makes the turbulence–mean shear interaction component of the mean-square wall pressure increase by a factor $2$ , within the assumptions of the simplified boundary-layer model and assuming that the length scales of the two turbulence models are equal. The same result was derived by Kraichnan (Reference Kraichnan1956a ) by integration of the wall-pressure correlation coefficient in the space domain. The study of a real boundary layer presented in § 4 will allow us to understand if this relationship between longitudinal correlation function and mean-square wall pressure holds also in the case of a real boundary layer.

3.3 Application to the turbulence–turbulence interaction component

Equation (2.42) can be integrated analytically in order to estimate the order of magnitude of the mean-square wall pressure due to the turbulence–turbulence source term of the Poisson equation. For illustrative purposes, the mean-square velocity $\overline{u^{2}}$ is considered constant in the $x_{2}$ direction. The result then depends on the choice of the ${\mathcal{F}}_{2}$ function. If the Gaussian function of (C 32) is substituted in (2.42) and the resulting formulation is integrated with respect to $X_{2}$ and $X_{2}^{\prime }$ , one obtains

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{pp}^{TT}(\boldsymbol{k})=\frac{\sqrt{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D70C}_{0}^{2}\,L^{3}\overline{u^{2}}^{2}}{512\,\unicode[STIX]{x03C0}}k\,(28+L^{2}k^{2})~\text{erfc}\left(\frac{\sqrt{2}}{4}kL\right).\end{eqnarray}$$

This result is obtained with a procedure similar to that described in § D.1. Then, the integration of (3.9) with respect to $\text{d}\boldsymbol{k}$ yields

(3.10) $$\begin{eqnarray}\langle p^{2}\rangle _{TT}=\frac{47}{30}\unicode[STIX]{x1D70C}_{0}^{2}\overline{u^{2}}^{2}\approx 1.57\unicode[STIX]{x1D70C}_{0}^{2}\overline{u^{2}}^{2}.\end{eqnarray}$$

If, on the contrary, the exponential function of (C 33) is substituted in (2.42), one obtains

(3.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{pp}^{TT}(\boldsymbol{k}) & = & \displaystyle \frac{\unicode[STIX]{x1D70C}_{0}^{2}\,L^{2}}{128\,\unicode[STIX]{x03C0}}\iint _{0}^{\infty }\overline{u^{2}}(X_{2})\,\overline{u^{2}}(X_{2}^{\prime })\,\text{e}^{-k(X_{2}+X_{2}^{\prime })}\,\text{e}^{-\left|X_{2}-X_{2}^{\prime }\right|/L}\,\text{d}X_{2}\,\text{d}X_{2}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \times \,k^{2}\,(28+L^{2}k^{2})\,\text{e}^{-L^{2}k^{2}/8}.\end{eqnarray}$$

In this case, the integral in $\text{d}X_{2}\,\text{d}X_{2}^{\prime }$ is solved by using the variable transformation $Y_{2}=X_{2}^{\prime }-X_{2}$ that yields

(3.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{0}^{\infty }\text{e}^{-kX_{2}}\,\text{d}X_{2}\int _{0}^{\infty }\text{e}^{-kX_{2}^{\prime }}\,\text{d}X_{2}^{\prime }~\text{e}^{-2\left|X_{2}-X_{2}^{\prime }\right|/L}\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{0}^{\infty }\text{e}^{-2kX_{2}}\,\left(\int _{-X_{2}}^{\infty }\text{e}^{-kY_{2}}\text{e}^{-2\left|Y_{2}\right|/L}\,\text{d}Y_{2}\right)\,\text{d}X_{2}=\frac{L}{k(Lk+2)}\end{eqnarray}$$

for $Lk\neq -2$ (which is always verified because $L$ and $k$ are positive by definition). Then, substituting equation (3.12) in (2.42) leads to

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{pp}^{TT}(\boldsymbol{k})=\frac{\unicode[STIX]{x1D70C}_{0}\,L^{3}\,\overline{u^{2}}^{2}}{128\,\unicode[STIX]{x03C0}}\frac{k\,(28+L^{2}k^{2})}{(Lk+2)}\,\text{e}^{-L^{2}k^{2}/8},\end{eqnarray}$$

which corresponds to Hodgson’s equation (B.16) except that the present coefficient $28$ appears as $22$ in the manuscript, which is most likely a typewriting error. Finally, the integration of (3.13) with respect to $\text{d}\boldsymbol{k}$ yields

(3.14) $$\begin{eqnarray}\langle p^{2}\rangle _{TT}\approx 1.22\,\unicode[STIX]{x1D70C}_{0}^{2}\overline{u^{2}}^{2}.\end{eqnarray}$$

These considerations based on a simplified boundary-layer model suggest that the shape of the vertical correlation function does not alter significantly the mean-square wall pressure for constant $\overline{u^{2}}$ . Again, the application of the turbulence–turbulence interaction theory to the DNS data presented in § 4.6 will allow us to assess the validity of this result in case of a real boundary layer. It can also be noticed that the turbulence–turbulence mean-square wall pressure does not depend on the integral length scale, unlike the analogous expressions derived for the turbulence–mean shear component in § 3.2.

Figure 3.

Distribution of wall-pressure sensors on the surface of the CD aerofoil.

4.1 Wall-pressure measurements

Measurements of the surface-pressure fluctuations on a controlled-diffusion (CD) aerofoil were performed in the anechoic open-jet wind tunnel at Université de Sherbrooke (see Padois et al. (Reference Padois, Laffay, Idier and Moreau2015) for an assessment of the capabilities of this facility for aeroacoustic investigations). The CD aerofoil is instrumented with remote microphone probes (RMP) on the suction side (see Moreau & Roger Reference Moreau and Roger2005). The RMPs are positioned along the streamwise and spanwise directions, as can be seen in figure 3. Each probe comprises a Knowles FG $23329$ P07 microphone remotely connected via a T junction to a pinhole of diameter of 0.5 mm on the suction side of the aerofoil and to a 2 m long anechoic termination on the other end to damp end reflections and avoid resonances (see for more details Perennes & Roger Reference Perennes and Roger1998). This microphone has a flat response in the frequency range of 100 to 10 kHz. For the calibration of the microphones a TMS microphone (130P10 ICP) with a diaphragm diameter equal to 6.35 mm was used. It is known to have a flat response in the frequency range of 100 to 20 kHz. The TMS microphone was calibrated using a Larson Davis 200 calibrator which has an uncertainty of approximately $\pm 0.2$ dB. The data were acquired using a NI 9234 module with a 24 bit resolution. The acquisition frequency was 25 600 Hz and the length of the signals was 30 s. Since the actual probes were not flush mounted but instead were placed at a remote location, this leads to attenuation of the surface pressure registered at the remote location. This attenuation can be attributed both to a change in section of the connecting tubes and the finite length of the tubing. Hence the actual signal measured by the remote microphone has an attenuated amplitude and is phase shifted. To account for these effects an in situ calibration procedure was realised.

Lastly, it should be kept in mind that, due to the finite size of the pinhole, some spatial averaging is anticipated especially at higher frequencies. Gravante et al. (Reference Gravante, Naguib, Wark and Nagib1998) performed a set of wall-pressure measurements with pinhole microphones of several diameters and showed that spectral attenuation can be avoided until $f^{+}=1$ if $12.0<d^{+}<18.0$ , where $f^{+}=f\,\unicode[STIX]{x1D708}_{T}/u_{\unicode[STIX]{x1D70F}}^{2}$ and $d^{+}=d\,u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}_{T}$ are the frequency of interest and diameter of the pinhole respectively, both normalised by wall units. It was also shown that the diameter of the pinhole has no influence on the mean-square wall pressure up to $d^{+}=27$ . Using the friction velocity calculated from the DNS, the $d^{+}$ for sensor $21$ was estimated to be equal to $26.0$ , whereas $d^{+}$ for sensor $24$ was estimated to be equal to $22.2$ . Likewise, $f^{+}=1$ corresponds to $f=39.1$ kHz for sensor 21 and to $f=28.6$ kHz for sensor 24. However, since the measurement range for the Knowles FG $23329$ P $07$ is restricted to $10$ kHz the attenuation effects are expected to be small. However, Roger (Reference Roger2017) has shown with analytical considerations that the attenuation due to integration effects at high frequencies depends on the statistical properties of the wall-pressure field to be measured. Then, the DNS-based wall-pressure statistics could be used in the future to understand better this effect.

4.2 DNS of the flow over the controlled-diffusion aerofoil

The three-dimensional DNS of the flow over the CD aerofoil is conducted using a multi-block structured high-order accurate compressible Navier–Stokes solver (Sandberg Reference Sandberg2015) taking the mean installation effects ( jet width) into account, as suggested by Moreau et al. (Reference Moreau, Henner, Iaccarino, Wang and Roger2003). The volume data around the aerofoil were recorded during the simulation at a sampling frequency of 78 kHz for 7 flow-through times based on the reference velocity and aerofoil chord length, which was found enough to yield properly converged flow statistics in previous large-eddy simulations (LES) and DNS on this aerofoil at this flow condition (see Wang et al. Reference Wang, Moreau, Iaccarino and Roger2009; Moreau et al. Reference Moreau, Sanjosé, Perot and Kim2011; Wu, Moreau & Sandberg Reference Wu, Moreau and Sandberg2019). A fourth-order central standard-difference scheme with Carpenter boundary stencils (Carpenter, Nordström & Gottlieb Reference Carpenter, Nordström and Gottlieb1999) is applied for the spatial discretisation in the streamwise and cross-wise directions. A spectral method using the FFTW3 library is used in the spanwise direction. Time marching is achieved by an ultra-low-storage five-step fourth-order Runge–Kutta scheme (Kennedy, Carpenter & Lewis Reference Kennedy, Carpenter and Lewis1999) with a constant time step of $\unicode[STIX]{x0394}t=6.4\times 10^{-8}$ s. Characteristic-based boundary conditions (described in Sandberg & Sandham Reference Sandberg and Sandham2006; Jones Reference Jones2008; Jones, Sandberg & Sandham Reference Jones, Sandberg and Sandham2008) (see also Kim & Lee Reference Kim and Lee2003) are used for this simulation on domain boundaries to avoid unphysical reflections (see for more details Wu et al. Reference Wu, Sanjosé, Moreau and Sandberg2018). An adiabatic, no-slip condition is set on the aerofoil surface. The grid refinement in the boundary layers is guided by the data obtained from experiments (Moreau et al. Reference Moreau, Henner, Iaccarino, Wang and Roger2003; Wu et al. Reference Wu, Laffay, Idier, Jaiswal, Sanjosé and Moreau2016). In the spanwise direction, 96 Fourier modes are employed with 100 % de-aliasing, which corresponds to 194 collocation points in physical space. These points are distributed with equal distance over a spanwise width of 12 % of the aerofoil chord length and are shown to resolve properly the Kolmogorov scale (see Wu et al. Reference Wu, Sanjosé, Moreau and Sandberg2018, Reference Wu, Moreau and Sandberg2019). The O-grid around the boundary layer comprises $3341\times 279\times 194$ grid points, which give finally a grid resolution of $\unicode[STIX]{x0394}x_{1,ave}^{+}\approx 6.0$ , $\unicode[STIX]{x0394}x_{2,ave}^{+}\approx 0.8$ and $\unicode[STIX]{x0394}x_{3,ave}^{+}\approx 5.5$ based on the friction velocity. These average values are calculated on the rear part of the aerofoil surface corresponding to 40 % of the aerofoil chord. Further details on the validation of the DNS can be found in Wu et al. (Reference Wu, Moreau and Sandberg2019) including turbulent kinetic energy budgets.

Figure 4.

(a) Comparison of the measured and DNS pressure coefficient, $C_{p}$ . (b) Clauser parameter, $\unicode[STIX]{x1D6FD}_{c}$ , calculated from the DNS results.

Table 2.

Parameters for the normalisation of frequency and wall-pressure PSD.

Mean velocity profiles extracted from the DNS data on the aerofoil surface and in the wake compare favourably with hot-wire measurements performed in the anechoic wind tunnel of the Université de Sherbrooke (UdeS) (see Wu et al. Reference Wu, Sanjosé, Moreau and Sandberg2018, Reference Wu, Moreau and Sandberg2019). The calculated pressure coefficient,

(4.1) $$\begin{eqnarray}C_{p}=\frac{p-p_{0}}{1/2\,\unicode[STIX]{x1D70C}_{0}\,U_{e}^{2}},\end{eqnarray}$$

where $p_{0}$ is the free-stream static pressure, is compared in figure 4(a) with wall-pressure measurements taken in the same wind tunnel, which has a free-stream turbulence intensity below 0.4 %. A laminar separation bubble can be observed at the leading edge, whose extension is well captured by the DNS. The onset of an adverse pressure gradient is visible on the rear half of the suction side. The development of the pressure gradient on the suction side of the aerofoil is presented in figure 4(b) using the Clauser parameter, defined as,

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{C}=\frac{\unicode[STIX]{x1D6FF}^{\ast }}{\unicode[STIX]{x1D70F}_{w}}\frac{\text{d}P}{\text{d}x_{1}}.\end{eqnarray}$$

The pressure gradient with respect to the abscissa along the aerofoil, $\text{d}P/\text{d}x_{1}$ , increases continuously from the mid-chord to the trailing edge.

The wall-pressure PSD calculated from the DNS results is compared in figure 5 with the measured one at three pressure sensors on the suction side of the aerofoil close to the trailing edge, namely sensor 21 (with $\unicode[STIX]{x1D6FD}_{C}=5.01$ ), sensor 24 (with $\unicode[STIX]{x1D6FD}_{C}=6.28$ ) and sensor 26 (with $\unicode[STIX]{x1D6FD}_{C}=9.72$ ). The boundary-layer parameters used for the normalisation of the curves are summarised in table 2. The boundary-layer thickness corresponds to the $x_{2}$ coordinate where the total pressure reaches 99 % of its maximum constant value along the wall-normal direction. The total pressure is used instead of the velocity because the aerofoil is loaded and therefore the external flow is deflected so that the external velocity is not constant. It can be noticed that at the lower frequencies the DNS spectrum is slightly below the experimental one. This is due to the fact that, whereas the simulation does not include the jet unsteadiness and acoustics, some significant effect from the jet is inevitable in the experiment, which prevents us obtaining the theoretical positive slope in the wall-pressure spectra. In the case of sensor 26, a high-frequency hump can be noticed, which is attributed to the contamination from the acoustic sources in the wake of the aerofoil. For this reason, the following analysis will focus on sensors 21 and 24 only.

Figure 5.

Comparison of the measured and DNS calculated wall-pressure PSD, $\unicode[STIX]{x1D711}_{pp}$ , on the suction side of the CD aerofoil corresponding to the locations of three pressure sensors.

Figure 6.

Normalised boundary-layer profiles extracted from the averaged DNS flow field corresponding to pressure sensors 21 and 24, respectively. (a) Mean streamwise velocity; (b) r.m.s. vertical velocity. The dotted lines on (a) indicate the locations chosen for the calculation of turbulence cross-correlation coefficients.

Finally, figure 6 presents the boundary-layer profiles of mean streamwise velocity and vertical r.m.s. fluctuation extracted from the DNS corresponding to sensors 21 and 24. Both plots are normalised in wall units. The effect of the increase of $\unicode[STIX]{x1D6FD}_{C}$ from sensors 21 to 24 can be observed in the mean velocity plot of figure 6(a), particularly in the wake zone of the boundary layer corresponding to $x_{2}^{+}>100$ . Also the vertical r.m.s. fluctuation of figure 6(b) is seen to increase with the Clauser parameter as expected.

4.3 Calculation of turbulence statistics

The DNS of the flow around the CD aerofoil is post-processed to gain information on the statistics of turbulence that are part of the above analytical model for the prediction of the wall-pressure spectrum. The wall-normal velocity correlation coefficient, $R_{22}$ , is analysed. In fact, the cross-spectral density of wall-normal velocity fluctuations, $\unicode[STIX]{x1D711}_{22}$ , which appears in (2.17), is the double spatial Fourier transform of $R_{22}$ (see (B 9)). The correlation coefficients calculated from the DNS results are compared with the analytical formulations that are derived from a given longitudinal correlation function, $F(r)$ , according to (B 6). This comparison assesses the accuracy of the analytical formulation of $\unicode[STIX]{x1D711}_{22}^{I}$ . The $R_{22}$ profiles discussed below are extracted from planes parallel to the suction side of the CD aerofoil using seven flow-through times for averaging. The results correspond to sensor 24. The same calculations performed for sensor 21 are not reported here for brevity as they yield equivalent results. The profiles are plotted versus the distance from the reference point, $r$ , normalised by the longitudinal integral length scale, $\unicode[STIX]{x1D6EC}$ , calculated at each wall-normal position. Furthermore, $R_{22}$ is calculated for several streamwise, $r_{1}$ , and transverse $r_{3}$ separations. The curves of $R_{22}(r_{1})$ and $R_{22}(r_{3})$ nearly collapse if the coordinate $r_{1}$ is divided by the coefficient $\unicode[STIX]{x1D6FC}$ that represents the ratio of streamwise to transverse longitudinal integral length scales (see (2.19)).

Figure 7.

Vertical velocity correlation coefficient calculated on a plane at $x_{2}^{+}=23$ in the streamwise ( $r_{1}$ ) and transverse ( $r_{3}$ ) directions. Ratio of streamwise to transverse length scales: $\unicode[STIX]{x1D6FC}=1.88$ .

Figure 7(a) shows the $R_{22}(r_{1})$ and $R_{22}(r_{3})$ profiles extracted from a plane at $x_{2}^{+}=23$ . In this case, it is sufficient to take $\unicode[STIX]{x1D6FC}=1.88$ for the curves to coincide, at least for small separations, $r$ . The transverse correlation $R_{22}(r_{3})$ is higher than $R_{22}(r_{1})$ for longer separations, whereas $R_{22}(r_{1})$ exhibits negative lobes. These facts point to the presence of coherent turbulent structures elongated in the transverse direction (see Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2014, figure 1). Wu et al. (Reference Wu, Sanjosé, Moreau and Sandberg2018) have shown that the onset of these structures is related to the increase of the adverse pressure gradient and that also the wall-pressure correlation coefficient increases in the transverse direction. In this case, all analytical formulations seem to overestimate the numerical correlation coefficients. Only the Liepmann formulation is in close agreement with the numerical curves for small values of $r$ . The von Kármán formulation is not shown in this analysis as its results are very close to that of Liepmann in all calculations.

Figure 8.

Vertical velocity correlation coefficient calculated on a plane at $x_{2}^{+}=65$ in the streamwise ( $r_{1}$ ) and transverse ( $r_{3}$ ) directions. Ratio of streamwise to transverse length scales: $\unicode[STIX]{x1D6FC}=1.55$ .

Figure 8 presents the same analysis of the velocity fluctuations extracted from a plane at $x_{2}^{+}=65$ . The elongation of the turbulence structures farther away from the wall is represented by the coefficient $\unicode[STIX]{x1D6FC}=1.55$ . In this case, the Gaussian formulation overestimates the negative undershoots of the numerical curves. On the contrary both the RDT and the Liepmann formulations give good approximations to the results. In particular, the Liepmann curve is closer to the computed $R_{22}$ for $r<1$ , whereas for $r>1$ the RDT curve is the closest.

Figure 9.

Vertical velocity correlation coefficient calculated on a plane at $x_{2}^{+}=165$ in the streamwise ( $r_{1}$ ) and transverse ( $r_{3}$ ) directions. Ratio of streamwise to transverse length scales: $\unicode[STIX]{x1D6FC}=1.35$ .

Figure 9 refers to a plane close to the outer edge of the boundary layer ( $x_{2}^{+}=165$ ), where the elongation of the turbulent structures is $\unicode[STIX]{x1D6FC}=1.35$ . It can be noticed that the Gaussian theory (figure 9 b) reproduces well the negative parts of the curve of $R_{22}$ . Figures 9(c) and 9(d) compare the DNS curves with the RDT and Liepmann formulations, respectively. In this case, the former analytical formulation provides a slightly better representation of the numerical results than the latter.

In the results presented so far it has been assumed that $\unicode[STIX]{x0394}x_{2}=0$ . In order to complete this analysis, the calculation of $R_{22}$ between two planes parallel to the wall is now presented. Figure 10 shows the correlation coefficient computed between the plane at $x_{2}^{+}=65$ and a second plane parallel to the wall which has a normalised distance $\unicode[STIX]{x0394}x_{2}/\unicode[STIX]{x1D6FF}=0.024$ from the first. Due to the wall-normal separation, the maximum value of $R_{22}$ , found at $r_{1}=r_{3}=0$ , is logically lower than unity. Visibly, the three analytical formulations predict differently the decay of $R_{22}$ with $\unicode[STIX]{x0394}x_{2}$ . In this case, the RDT and Liepmann formulations are in better agreement with the numerical data than the Gaussian. Finally, figure 11 presents the $R_{22}$ coefficient calculated between the plane at $x_{2}^{+}=65$ and another plane at $\unicode[STIX]{x0394}x_{2}/\unicode[STIX]{x1D6FF}=0.05$ from the first. It can be noticed that the peak of $R_{22}(r_{1})$ corresponds to a $r_{1}$ slightly below zero, whereas the plot of $R_{22}(r_{3})$ is centred on $r_{3}=0$ . This is due to the fact that the velocity time series has been extracted directly at the mesh cell centres, in order to avoid the numerical error introduced by interpolation. The mesh follows the curvature of the aerofoil surface, so the cell centres are not perfectly aligned in the wall-normal direction, whereas there is perfect alignment in the transverse direction. It must be emphasised that if the fundamental assumption of the TNO-Blake model, expressed in (2.31), was true, then the DNS-based statistics presented in figures 10 and 11 would vanish for any $r$ .

The decay of the maximum $R_{22}$ with the increase of the vertical separation is plotted in figure 12. The reference point for this calculation has been taken at roughly half the boundary layer thickness for both sensors 21 and 24. The second point for the calculation of the cross-correlation coefficient has been chosen moving upwards in the vertical direction. As will be shown below, the longitudinal integral length scale of turbulence is nearly constant in this region of the boundary layer. For this reason, it has been possible to normalise the abscissa of figure 12 using the averaged $\unicode[STIX]{x1D6EC}$ in the outer part of the boundary layer. It can be observed that, especially for small separations, the RDT is in best agreement with the DNS data corresponding to both sensors. For higher vertical separations, the DNS data are between the RDT and Liepmann theoretical curves. The cross-correlation coefficient of vertical velocity fluctuations is significantly higher than zero on a vertical extension corresponding to two integral length scales, thus further contradicting equation (2.31).

The general conclusion that can be drawn from this study is that especially the RDT and, second, the Liepmann models provide an acceptable description of the turbulence statistics in this adverse-pressure-gradient boundary layer. The present result is also very consistent with what Magnaudet (Reference Magnaudet2003) found: the rapid distortion theory is the leading-order approximation capable of describing short- and long-time evolutions of turbulent boundary layers in the limit of large Reynolds number. In any case, all analytical formulations of the cross-spectral density of wall-normal turbulent fluctuations, $\unicode[STIX]{x1D711}_{22}$ , that have been presented in § 2.3 will be applied to the prediction of the wall-pressure spectra on the surface of the CD aerofoil in § 4.5. In this way, the relationship between the shape of turbulence statistics and that of the corresponding wall-pressure PSD will be assessed.

Figure 10.

Vertical velocity correlation coefficient calculated between a reference plane at $x_{2}^{+}=65$ and a second plane parallel to the wall at $\unicode[STIX]{x0394}x_{2}/\unicode[STIX]{x1D6FF}=0.024$ from the first. Ratio of streamwise to transverse length scales: $\unicode[STIX]{x1D6FC}=1.55$ .

Figure 11.

Vertical velocity correlation coefficient calculated between a reference plane at $x_{2}^{+}=65$ and a second plane parallel to the wall at $\unicode[STIX]{x0394}x_{2}/\unicode[STIX]{x1D6FF}=0.05$ from the first. Ratio of streamwise to transverse length scales: $\unicode[STIX]{x1D6FC}=1.55$ .

Figure 12.

Evolution of the maximum vertical velocity correlation coefficient, $R_{22}$ , with the vertical separation. Comparison of the DNS-based result with the analytical formulations.

In order to compute $\unicode[STIX]{x1D711}_{22}$ with any analytical formulation, it is necessary to know the variation of the length scale $\unicode[STIX]{x1D6EC}$ with the distance from the wall. This is depicted in figure 13(a) corresponding to sensors 21 and 24, for which the wall-pressure spectrum will be calculated. Since $\unicode[STIX]{x1D6EC}$ is a quantity that is defined in the theory of homogeneous turbulence (see, for instance Wilson (Reference Wilson1997, Reference Wilson1998) and appendix B), it should only be calculated with respect to spatial directions in which the turbulence can be, to a good approximation, considered homogeneous. In this case, the correlation of the streamwise velocity component, $R_{11}$ , has been calculated as a function of the streamwise separation, $r_{1}$ , on planes at given distances from the wall. According to (B 3), $\unicode[STIX]{x1D6EC}$ is the integral of this correlation function. Since the tail of the function tends to be noisy, similarly to the $R_{22}$ functions previously examined, $R_{11}$ has been calculated over several parallel streamwise lines and the results have been averaged in order to obtain a smooth curve allowing a robust calculation of the integral. Interestingly, the plateau of $\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D6FF}$ corresponds to a higher value than the classical $0.14$ of Prandtl theory (see Panton & Linebarger Reference Panton and Linebarger1974). Figure 13(b) shows the trend of $\unicode[STIX]{x1D6FC}$ along the wall-normal direction: the length scales are equal ( $\unicode[STIX]{x1D6FC}=1$ ) for $x_{2}/\unicode[STIX]{x1D6FF}>0.8$ corresponding to both pressure sensors. Then, $\unicode[STIX]{x1D6FC}$ slowly rises to $2.0$ corresponding to $x_{2}/\unicode[STIX]{x1D6FF}=0.2$ and reaches the value $3.0$ at the inner edge of the boundary layer, but only for sensor 24. Instead, the maximum $\unicode[STIX]{x1D6FC}$ equals $2.5$ corresponding to sensor 21. Yet, very similar variations are observed for these two different adverse-pressure-gradient conditions.

Figure 13.

(a) Longitudinal integral length scale profiles calculated for the streamwise velocity component. (b) Ratio of the streamwise to transverse integral length scales, $\unicode[STIX]{x1D6FC}$ , corresponding to sensor 24.

4.4 Estimation of the convective speed of wall-pressure fluctuations

The wavenumber wall-pressure spectra predicted with (2.17) will be converted into the frequency domain by using (2.18) for the sake of comparison with the measured and directly computed spectra. For this purpose, it is necessary to define the value of the convective speed of the wall-pressure fluctuations, $U_{c}$ . This quantity is, in principle, a function of the frequency. The empirical model of $U_{c}$ developed by Smol’yakov (Reference Smol’yakov2006) for zero-pressure-gradient boundary layers, however, predicts that this quantity will tend to a constant value with the increase of $\unicode[STIX]{x1D714}$ . Salze et al. (Reference Salze, Bailly, Marsden, Jondeau and Juve2014) have shown that this trend is also valid for adverse and favourable pressure gradients. The reference measured and computed spectra correspond, in the present application case, to a range of frequencies where, according to the results presented by these authors, $U_{c}$ can be considered to a good approximation a constant fraction of the external flow speed, $U_{0}$ . The estimation of $U_{c}$ is made by means of the cross-correlation of two pressure signals extracted from the DNS over seven flow-through times at a sampling frequency of 78 663 Hz. This has been done for both sensors 21 and 24. The two points on the surface of the aerofoil are aligned in the transverse direction and are separated by a given distance in the streamwise direction. The points around sensor 21 are separated by $0.85\unicode[STIX]{x1D6FF}$ , whereas those around sensor 24 are separated by $\unicode[STIX]{x1D6FF}$ . Since the largest size of the vortical structures corresponds roughly to the boundary-layer thickness, pressure signals recorded at locations more than $\unicode[STIX]{x1D6FF}$ apart are expected to be statistically uncorrelated. Figure 14 depicts the wall-pressure cross-correlations versus the time lag, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}$ . The maximum cross-correlation is indicated with a circle in each plot. The distance between the two points where the pressure signals were recorded divided by the time lag corresponding to the maximum cross-correlation gives an estimate of the convection speed. It is found that for sensor 21, $U_{e}/U_{c}=1.51$ , whereas for sensor 24, $U_{e}/U_{c}=1.54$ . These values are consistent with those found in the previous experimental investigations of Roger & Moreau (Reference Roger and Moreau2004) and Moreau & Roger (Reference Moreau and Roger2005) on the CD aerofoil.

Figure 14.

Wall-pressure normalised correlation versus time lag.

Figure 15.

Comparison between wall-pressure PSDs computed predicted with RDT assuming $\unicode[STIX]{x1D6FC}=f(x_{2})$ and $\unicode[STIX]{x1D6FC}=1$ . Black: sensor 21. Grey: sensor 24.

4.5 Prediction of the wall-pressure PSD

The wall-pressure PSD is predicted by using the mean shear, $\unicode[STIX]{x2202}U_{1}/\unicode[STIX]{x2202}x_{2}$ , and r.m.s. of the wall-normal velocity fluctuations, $\overline{u_{2}}$ , as input to (2.17) and (2.18). The mean and r.m.s. velocity profiles have been presented in figure 6. The effect of the anisotropy of the length scales on the predicted wall-pressure PSD can be verified, as a preliminary test, by choosing one wall cross-spectral density formulation and performing the calculation of $\unicode[STIX]{x1D711}_{pp}$ twice. The first calculation is made taking into account in (2.19) the variation of $\unicode[STIX]{x1D6FC}$ with $x_{2}$ (depicted in figure 13). The second calculation is performed with $\unicode[STIX]{x1D6FC}=1$ . The results of this test are shown in figure 15 for sensors 21 and 24 using the RDT turbulence model. It is then shown that the wall-pressure PSD in the frequency range of interest is determined by the isotropic part of the boundary layer. As a matter of fact, the integrand of (2.17) is directly proportional to $\unicode[STIX]{x2202}U_{1}/\unicode[STIX]{x2202}x_{2}$ and $\overline{u_{2}}$ , multiplied by an exponential function that decays proportionally to the product of the distance from the wall by the magnitude of the wavenumber vector. Consequently, it is possible that the wall-pressure PSD is mostly determined by the near-wall physics of the boundary layer, as suggested by the study of Panton & Linebarger (Reference Panton and Linebarger1974). The outer part of the boundary layer is expected to play an important role only at very low frequencies, which correspond to small $k_{1}$ values. However, since a comparison with measured and DNS wall-pressure PSD can be made only in the medium–low-frequency range, the following computations will be performed assuming $\unicode[STIX]{x1D6FC}=1$ without any loss of accuracy in the range of frequencies of interest.

Since all relevant physical quantities are known from the DNS, the prediction of wall-pressure PSD can be used to assess the effect of the different analytical formulations of $\unicode[STIX]{x1D711}_{22}$ presented in § 2.3. The results for sensors 21 and 24 are presented in figure 16. Comparing the results in figures 16 and 1 clearly highlights the relationship between the longitudinal correlation function of turbulence and the corresponding predicted wall-pressure PSD. The flattening of $F(r)$ from the Gaussian to the von Kármán correlation function, indicated by the arrows in figure 1, corresponds to a marked increase of the wall-pressure PSD in the middle-frequency range and a slight decrease at high frequencies, also indicated with the arrows in figure 16. Overall, the flattening of the turbulence correlation function corresponds to an increase of the mean-square pressure at the wall, a result that has been foreseen by means of purely analytical considerations in § 3.2. It can be noticed that, for $\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D6FF}/U_{e}>5$ , the RDT and Gaussian $\unicode[STIX]{x1D711}_{22}$ expressions provide the best agreement with the directly computed curve, whereas in the lower-frequency range the Liepmann and RDT-based curves are much closer to the DNS. The good agreement between the DNS and RDT curves is particularly visible in the sensor 24 plot, whereas for the sensor 21 the Liepmann model performs slightly better at low frequencies. As a general conclusion, considering that the difference between the various formulations at high frequencies is relatively small, the RDT and possibly Liepmann turbulence spectrum provide a proper description over the whole frequency range that has been considered ( $0.15<\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D6FF}/U_{e}<25$ ). This is consistent with the considerations made on the turbulence statistics earlier in this section. Furthermore, the use of the turbulence–mean shear interaction source term alone is sufficient to retrieve the correct level of the wall-pressure PSD on the explored range of frequencies. However it is interesting to apply also the turbulence–turbulence interaction formulation in order to estimate its contribution to the overall spectrum.

Figure 16.

Comparison between wall-pressure PSDs computed directly and predicted with different assumptions on the shape of the longitudinal correlation function.

4.6 Application of the turbulence–turbulence interaction component formulation

The formulation presented in § 2.4 can be applied to the prediction of the turbulence–turbulence interaction wall-pressure PSD in the same way as the turbulence–mean shear formulation. The results obtained using both ${\mathcal{F}}_{2}$ formulations are presented in figure 17. It can be seen that the curve obtained with the Gaussian vertical correlation function is only slightly higher than the curve obtained with the exponential ${\mathcal{F}}_{2}$ formulation, as suggested by the previous analytical considerations. These numerical results suggest that the turbulence–turbulence source is significant only at high frequencies but thoroughly underestimates the main low-frequency contribution to the overall mean-square pressure. The turbulence-turbulence (TT) curves presented in figure 17 exhibit a plateau for $\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}/U_{e}<5$ , a trend that can also be observed in figure 17 of Slama et al. (Reference Slama, Leblond and Sagaut2018). However, this conclusion must be taken with caution since it is obtained by making such a critical hypothesis as that of a normal probability distribution and isotropic turbulence. This should be further evaluated in the DNS results in the future.

Figure 17.

Comparison of the turbulence–turbulence interaction component calculated using two different vertical correlation functions. The DNS wall-pressure PSD is shown for reference.

Figure 18.

Comparison of the r.m.s. wall-normal velocity fluctuations extracted from DNS and modelled according to the TNO-Blake approach.

4.7 Application of the TNO-Blake model

The TNO-Blake model can be applied to the DNS data in order to assess the effect of the substitution of the turbulence cross-spectrum with the single-point auto-spectrum and of the hypothesis on the boundary-layer profiles of $\overline{u_{2}}$ and $\unicode[STIX]{x1D6EC}$ on the predicted wall-pressure PSD.

First, the empirical equations used in the original TNO-Blake model can be compared with the boundary-layer profiles extracted from the DNS on the CD aerofoil corresponding to the sensors $21$ and $24$ . Figure 18 compares the directly computed $\overline{u_{2}}$ with the results of the application of (2.36) to the mean-shear profiles. It appears from these results that the empirical formulation fails to reproduce the DNS curve. The small local peaks in the modelled $\overline{u_{2}}$ curve are due to the numerical calculation of the velocity gradient of the mean flow interpolated on a line perpendicular to the blade surface. On the contrary, it is shown in figure 19 that (2.39) provides a slightly underestimated approximation of the computed $\unicode[STIX]{x1D6EC}$ .

Figure 19.

Comparison of the longitudinal integral length scale of turbulence extracted from DNS and modelled according to the TNO-Blake approach.

Figure 20.

Comparison of the wall-pressure PSD computed from DNS and using the TNO-Blake turbulence model with the modelled boundary-layer profiles as input.

The accuracy of the TNO-Blake model in the prediction of the wall-pressure PSD can be evaluated in two steps. First, equation (2.32) is computed using the profiles of $\overline{u_{2}^{2}}$ and $\unicode[STIX]{x1D6EC}$ given by the empirical equations (2.36) and (2.39), respectively. Second, and perhaps more interestingly, equation (2.32) is computed using the profiles of $\overline{u_{2}}$ and $\unicode[STIX]{x1D6EC}$ extracted from the DNS of the controlled-diffusion aerofoil, as done in § 4.5. Both sets of calculations are performed assuming isotropy of the length scales ( $\unicode[STIX]{x1D6FC}=1$ ), following the results presented in § 4.5.

In the first test the modelled wall-pressure PSD is underestimated with respect to the DNS result, as can be seen in figure 20, and this is a direct consequence of the inaccurate representation of boundary-layer profiles. The results of the second test are depicted in figure 21. In this case, the wall-pressure PSD obtained with the exact expression of the von Kármán cross-spectral density, already presented in § 4.5, has been reproduced for comparison. It can be seen that the approximate solution of the TNO-Blake model and the one based on the exact formulation of the turbulence cross-spectral density yield similar results in terms of wall-pressure PSD at high frequency but diverge at low frequency. In the range $\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D6FF}/U_{e}<5$ the wall-pressure PSD predicted using the vertical velocity auto-spectrum underestimates the prediction obtained with the exact formulation of the cross-spectrum but, thanks to this very fact, it also gets closer to the DNS result. Indeed, in § 4.5, the von Kármán theory was shown to overestimate the wall-pressure PSD at low frequencies. In the TNO-Blake model, this inaccuracy is compensated by substituting the cross-spectral density with the single-point auto-spectrum, although the latter assumption is quite wrong as shown in § 4.3.

Figure 21.

Comparison of the wall-pressure PSD computed from DNS and using the TNO-Blake turbulence model with the DNS-extracted boundary-layer profiles as input.

Section 2.3.3 has illustrated the treatment of the anisotropy of length scales by means of the coefficients presented in different versions of the TNO-Blake model (see table 1). The final test of this section concerns the effect that these different sets of coefficients have on the predicted wall-pressure PSD. The results, obtained for sensors 21 and 24, are depicted in figure 22, where the DNS spectrum and the TNO-Blake prediction obtained with the hypothesis of isotropy of the length scales are reported for comparison. The earliest set of coefficients proposed by Bertagnolio et al. (Reference Bertagnolio, Fischer and Zhu2014) significantly overestimates the DNS result at low frequencies. However, the later sets of coefficients proposed by Stalnov et al. (Reference Stalnov, Paruchuri and Joseph2016) and Fischer et al. (Reference Fischer, Bertagnolio and Madsen2017) are more conservative in the sense that they do not vary significantly the prediction with respect to the isotropic case.

Figure 22.

Comparison of the wall-pressure PSD computed from DNS and using the TNO-Blake turbulence model with the DNS-extracted boundary-layer profiles as input and using different sets of $\unicode[STIX]{x1D6FD}_{i}$ coefficients to account for turbulence anisotropy.

5.1 Recursive stratified sampling

This method has been developed by Press & Farrar (Reference Press and Farrar1990) and is implemented in the widely used MISER algorithm. In order to understand its advantages, it is necessary to take a step back and introduce the simplest form of Monte Carlo integration: the quasi-random sampling. Given a generic function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ , sampled with $N$ random vectors, $\mathbf{x}_{i}$ , uniformly distributed over the integration domain, the function mean estimator is defined as

(5.1) $$\begin{eqnarray}\langle \,\tilde{f}\rangle =\frac{1}{N}\mathop{\sum }_{i=1}^{N}f(\mathbf{x}_{i}).\end{eqnarray}$$

Then, the estimator of the integral is the product of $\langle \,\tilde{f}\rangle$ by the size of the integration volume. It can be shown (see James Reference James1980) that the variance of the estimator is related to the variance of the function over the integration domain as

(5.2) $$\begin{eqnarray}\text{Var}(\langle \,\tilde{f}\rangle )=\frac{\text{Var}(f)}{N}\end{eqnarray}$$

and that the following asymptotic relation holds for the error of the estimation

(5.3) $$\begin{eqnarray}\unicode[STIX]{x1D716}=O\left(\sqrt{\frac{\text{Var}(\langle \tilde{f}\rangle )}{N}}\right).\end{eqnarray}$$

Thus, the convergence of the quasi-random sampling is independent of the number of dimensions of integration. The recursive stratified sampling differs from quasi-random sampling because it divides the integration region $V$ into two equal and disjoint sub-volumes, $V_{a}$ and $V_{b}$ . The sub-volumes are then sampled at $N_{a}$ and $N_{b}=N-N_{a}$ random points, respectively, drawn from a uniform probability distribution function. In this case, the function estimator is defined as

(5.4) $$\begin{eqnarray}\langle \tilde{f}\rangle ^{\prime }={\textstyle \frac{1}{2}}(\langle \tilde{f}\rangle _{a}+\langle \tilde{f}\rangle _{b}),\end{eqnarray}$$

where the $\langle \tilde{f}\rangle _{a,b}$ terms are computed by applying equation (5.1) to each sub-volume. It can also be shown that the variance of $\langle \tilde{f}\rangle ^{\prime }$ is

(5.5) $$\begin{eqnarray}\text{Var}(\langle \tilde{f}\rangle ^{\prime })=\frac{1}{4}\left(\frac{\text{Var}_{a}(f)}{N_{a}}+\frac{\text{Var}_{b}(f)}{N-N_{a}}\right),\end{eqnarray}$$

which is minimised when

(5.6) $$\begin{eqnarray}\frac{N_{a}}{N}=\frac{\unicode[STIX]{x1D70E}_{a}}{\unicode[STIX]{x1D70E}_{a}+\unicode[STIX]{x1D70E}_{b}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}=\sqrt{\text{Var}(\langle \tilde{f}\rangle )}$ . If $N_{a}$ satisfies equation (5.6), the variance of the function mean estimator is

(5.7) $$\begin{eqnarray}\text{Var}(\langle \tilde{f}\rangle ^{\prime })=\frac{\left(\unicode[STIX]{x1D70E}_{a}+\unicode[STIX]{x1D70E}_{b}\right)^{2}}{4N}.\end{eqnarray}$$

The advantage of this method is that $\text{Var}(\langle \tilde{f}\rangle ^{\prime })$ as defined in (5.7) is never greater than the variance of the estimator computed with the quasi-random sampling, defined in (5.2). In practice, a fixed number of function evaluations $N$ is allocated at the beginning of the execution of the algorithm. Then each dimension of integration is divided in two disjoint sub-volumes. A given fraction of the $N$ samples is used to estimate the variance of the function on each sub-volume, so that the remaining points can be allocated according to (5.6). In order to converge to a more accurate estimation, the algorithm can be applied recursively on the sub-volumes where the integrand function has greater variance.

5.2 Application to the turbulence–mean shear component of the wall-pressure spectrum

As mentioned at the beginning of this section, the expression of the turbulence–mean shear component of the wall-pressure PSD resulting from the substitution of (2.17) in (2.18) is a three-dimensional (3-D) integral which can be calculated numerically with quadratures in a reasonable computational time. However, the same turbulence–mean shear component can be expressed by a four-dimensional (4-D) integral if the one-dimensional integral expression of $\unicode[STIX]{x1D711}_{22}$ given in (B 13) is substituted in (2.17). This expression can also be made five-dimensional (5-D) if the two-dimensional (2-D) integral expression of $\unicode[STIX]{x1D711}_{22}$ given in (B 10) is used instead. Thus, this problem in which the same physical quantity can be expressed by a 3-D, 4-D or 5-D integral formulation is particularly suited for the evaluation of the performance of a Monte Carlo method. The recursive stratified sampling is applied to the calculation of the turbulence–mean shear wall-pressure PSD, with the 4-D and 5-D formulations, corresponding to sensor 24 of the CD aerofoil using the boundary-layer profiles presented in § 4. The von Kármán model of turbulence statistics has been selected for this study. Analogous tests have been presented in Grasso et al. (Reference Grasso, Jaiswal and Moreau2018) using the Gaussian turbulence model, which has a simpler mathematical formulation. The MISER algorithm used to obtain the following results is available in the Scikit-Monaco package for Python (see Buignon Reference Buignon2013).

Figure 23 presents the convergence of the 4-D and 5-D formulations of the wall-pressure PSD calculated with the MISER algorithm for an increasing number of sample points. The spectra have been computed with $N$ ranging from $2000$ to $2\times 10^{5}$ points. It can be seen that the integration is challenging from the numerical point of view only at the highest and lowest frequencies. However, starting from $2\times 10^{4}$ sampling points, all curves coincide. It is thus shown that the convergence of the MISER algorithm depends on the number of sampling points but not on the number of dimensions of integration, as postulated by the theory. The calculation of the wall-pressure PSD at $25$ values of $\unicode[STIX]{x1D714}$ , using $2\times 10^{4}$ sampling points for each frequency, has been performed in parallel on $4$ cores of an Intel Xeon Processor EG-1630 v3 with base frequency of 3.70 GHz. The computational time is $8.6$ s for the 3-D formulation and $10.2$ s for both the 4-D and the 5-D integral formulations. Increasing the number of sampling points for each frequency of one order of magnitude, that is $2\times 10^{5}$ , the computational time corresponding to $25$ values of $\unicode[STIX]{x1D714}$ integrated in parallel on $4$ cores rises to $1$ min for the 3-D formulation and to $1$ min $13$ s for both the 4-D and the 5-D integral formulations. Finally, figure 24 presents the somewhat obvious but necessary verification that the integration algorithm converges to the same prediction of the wall-pressure PSD using the 3-D, 4-D and 5-D formulations, which are equivalent from the analytical point of view.

This application shows that the MISER algorithm can be recommended for the prediction of wall-pressure PSD due to its fast and robust convergence, the ease of implementation and the absence of conditions on the shape of the integrand function.

Figure 23.

Convergence of the Monte Carlo integration of the wall-pressure spectrum 4-D and 5-D formulations.

Figure 24.

Correspondence of the Monte Carlo integration of the wall-pressure spectrum 4-D and 5-D formulations with the 3-D formulation.