3.1. One-dimensional spectra

We start our results analyses with the one-dimensional spectra. We note that the one-dimensional spectra of the wall pressure fluctuation have been previously studied (Abe et al. Reference Abe, Matsuo and Kawanura2005; Anantharamu & Mahesh Reference Anantharamu and Mahesh2020). The purpose of presenting the results of the one-dimensional spectra is to provide a further validation of our DNS data.

Figure 2 compares the one-dimensional streamwise-wavenumber spectrum, spanwise- wavenumber spectrum and frequency spectrum of the wall pressure fluctuation at various Reynolds numbers. The spectra are non-dimensionalized using the wall units. To validate our results, the one-dimensional spectra of wall pressure fluctuations at ${Re}_{\tau} =180$ calculated by Choi & Moin (Reference Choi and Moin1990) are superposed. The present results for case CH180 are in good agreement with the results of Choi & Moin (Reference Choi and Moin1990). All of the one-dimensional spectra at different Reynolds numbers are close to each other at high wavenumbers and high frequencies, indicating that the spectra are scaled by the wall units (Farabee & Casarella Reference Farabee and Casarella1991; Hu et al. Reference Hu, Morfey and Sandham2006; Hwang et al. Reference Hwang, Bonness and Hambric2009). At small wavenumbers or frequencies, the magnitudes of all one-dimensional spectra increase monotonically with the Reynolds number, an observation that agrees with the DNS results of Abe et al. (Reference Abe, Matsuo and Kawanura2005). Figures 2(a) and 2(c) show that as the streamwise wavenumber $k_x$ or frequency $\omega$ increases, the magnitude of the corresponding one-dimensional spectrum first increases, and then decreases. However, as depicted in figure 2(b), the magnitude of the one-dimensional spanwise-wavenumber spectrum $\phi _{pp}(k_z)$ decreases monotonically as $k_z$ increases.

Figure 2.

One-dimensional spectra of wall pressure fluctuations at ${Re}_{\tau} =179$, 333, 551 and 998 with respect to (a) streamwise-wavenumber, (b) spanwise-wavenumber and (c) frequency. All spectra are non-dimensionalized using wall units. The results of Choi & Moin (Reference Choi and Moin1990) at ${Re}_{\tau} =180$ are shown for comparison.

3.2. Two-dimensional spectra

In this subsection, we analyse the two-dimensional spectra. To keep the paper concise, we focus on the results at ${Re}_{\tau} =998$. Figures 3(a) and 3(b) show the contours of the two-dimensional $k_x$–$\omega$ and $k_x$–$k_z$ spectra of the wall pressure fluctuation $\phi _{pp}$, respectively. For comparison, the contours of the $k_x$–$\omega$ and $k_x$–$k_z$ spectra of the streamwise velocity fluctuation $\phi _{uu}$ at $y^+=15$ are plotted in figures 3(c) and 3(d), respectively. We note that the $k_x$–$\omega$ spectra are symmetric about $(k_x,\omega ) = (0,0)$ and the $k_x$–$k_z$ spectra are symmetric about both $k_x=0$ and $k_z=0$. Therefore, only spectra at $k_x\ge 0$ are plotted in figure 3. As shown in figures 3(a) and 3(c), large magnitudes of $\phi _{pp}(k_x,\omega )$ and $\phi _{uu}(k_x,\omega )$ occur around their convection lines $\omega /k_x=U_{bc}$, demarcated using the dash–dotted lines. Here, the bulk convection velocity $U_{bc}$ is defined as (del Álamo & Jiménez Reference del Álamo and Jiménez2009)

(3.1)\begin{equation} {U_{bc}} = \frac{{\int {\int {{k_x}\omega \phi ({k_x},\omega )\,{\rm d}\omega \,{\rm d}{k_x}} } }}{{\int {\int {k_x^2\phi ({k_x},\omega )\,{\rm d}\omega \,{\rm d}{k_x}} } }}, \end{equation}

where $\phi (k_x,\omega )$ represents the $k_x$–$\omega$ spectrum of either wall pressure fluctuations or velocity fluctuations. The values of the bulk convection velocity for wall pressure fluctuations ($u_{bc}^{+}=11.6$) and streamwise velocity fluctuations at $y^+=15$ ($u_{bc}^{+}=10.7$) are close to each other, indicating that the convection property of wall pressure fluctuations is similar to the velocity fluctuations in the buffer layer. This is consistent with the conclusion of Anantharamu & Mahesh (Reference Anantharamu and Mahesh2020) that the velocity gradients in the buffer layer are the dominant sources of wall pressure fluctuations.

Figure 3.

Isopleths of two-dimensional spectra of wall pressure fluctuations at $y^+ = 0$ and streamwise velocity fluctuations at $y^+=15$. (a) The $k_x$–$\omega$ spectrum of wall pressure fluctuations $\phi _{pp}(k_x,\omega )$; (b) $k_x$–$k_z$ spectrum of wall pressure fluctuations $\phi _{pp}(k_x,k_z)$; (c) $k_x$–$\omega$ spectrum of streamwise velocity fluctuations $\phi _{uu}(k_x,\omega )$; and (d) $k_x$–$k_z$ spectrum of streamwise velocity fluctuations $\phi _{uu}(k_x,k_z)$. The Reynolds number is ${Re}_{\tau} =998$.

Another important observation from figure 3(a) is that as $k_x$ approaches zero, the isopleths gradually contract towards $(k_x,\omega )=(0,0)$, especially at the energy-containing scales (see the two innermost isopleths of $\phi _{pp}(k_x,\omega )=10^{-4}$ and $10^{-5}$). This contracting feature is not observed from the isopleths of $\phi _{uu}(k_x,\omega )$, which are approximately parallel to the convection line at low streamwise wavenumbers. Similar contracting isopleths also appear in the $k_x$–$k_z$ spectrum of wall pressure fluctuations $\phi _{pp}(k_x,k_z)$ as shown in figure 3(b), but are absent for $\phi _{uu}(k_x,k_z)$ in figure 3(d). This contracting behaviour suggests that as $k_x$ increases from zero, the spectrum of wall pressure fluctuations first increases in a small range before it decreases. Such a non-monotonic behaviour is consistent with the one-dimensional spectrum $\phi _{pp}(k_x)$ plotted in figure 2(a). Later in § 5.3, we will further show that this contracting behaviour is correlated to the rapid pressure fluctuations.

3.3. The three-dimensional spectrum

In this section, we continue to analyse the three-dimensional spectrum $\phi _{pp}(k_x,k_z,\omega )$ of wall pressure fluctuations at $Re_\tau =998$. The variation of $\phi _{pp}(k_x,k_z,\omega )$ with respect to $k_x$ corresponding to $k_z=0$ and five frequencies $\omega h/u_\tau =123$, 245, 409, 573 and 716 is plotted in figure 4(a), while figure 4(b) shows the variation of $\phi _{pp}(k_x,k_z,\omega )$ as a function of $k_z$ for $k_x=0$ and the same five frequencies. Convective peaks (denoted by the filled circles) can be identified in all of the five lines in figure 4(a). As the frequency $\omega$ increases, the location of the convective peak moves to a higher streamwise wavenumber. Focusing on each profile, as the streamwise wavenumber $k_x$ decreases from the convective wavenumber to zero, the value of the $\phi _{pp}(k_x,k_z,\omega )$ first decreases, and then increases slightly at low streamwise wavenumbers, forming a weak local maximum at the lowest resolved streamwise wavenumber. This local maximum is identified as an artificial acoustic mode by Choi & Moin (Reference Choi and Moin1990), which is a numerical artefact induced by the periodic boundary condition imposed in the streamwise direction. It is seen from figure 4(a) that for $\omega h/u_\tau =123$, the artificial acoustic mode only appears at the lowest resolved streamwise wavenumber $k_xh=0.5$. The range of the streamwise wavenumber influenced by the artificial acoustic mode becomes larger as the frequency increases, but is still confined in a small neighbouring region around the lowest resolved streamwise wavenumber. Similarly, artificial acoustic modes also occur in the spanwise direction but are confined to small spanwise wavenumbers as shown in figure 4(b).

Figure 4.

Three-dimensional wavenumber–frequency spectrum of wall pressure fluctuations at specific wavenumbers and frequencies: (a) $k_z=0$ and (b) $k_x=0$. The five lines in (a,b) correspond to five frequencies $\omega h/u_\tau =123$, 245, 409, 573 and 716. The arrow points to the increasing direction of the frequency. The convective peaks are marked using filled circles in (a). The Reynolds number is $Re_\tau = 998$.

Figures 5(a) and 5(b) show the contours of $\phi _{pp}(k_x, k_z, \omega )$ corresponding to $\omega h/u_\tau =245$ and 716, respectively. Convective peaks can be identified for both frequencies (denoted by the cross symbols in figure 5), and the spectral value decays as the wavenumber moves away from the convective peaks. The isopleths are closed, ellipse-like curves, except for a small region near $(k_x,k_z)=(0,0)$. Owing to the artificial acoustic modes, the value of $\phi _{pp}(k_x, k_z, \omega )$ obtained from DNS does not decay to zero at $(k_x,k_z)=(0,0)$. To facilitate a clearer observation near $(k_x,k_z)=(0,0)$, figure 5(c) displays a zoom-in view of the dash–dotted box in figure 5(a). It is seen that the artificial acoustic modes only influence a small region around $(k_x,k_z)=(0,0)$ (marked by the red circle in figure 5c). Outside this circle, the isopleths form a ‘valley’ between $(k_x,k_z)=(0,0)$ and the convective peak. This observation shows the tendency that the magnitude of $\phi _{pp}(\boldsymbol {k},\omega )$ decays as the spanwise wavenumber approaches $k_z=0$ in the $k_x$–$k_z$ plane, which is in agreement with figure 4(b).

Figure 5.

Contours of the three-dimensional wavenumber–frequency spectrum of wall pressure fluctuations $\phi _{pp}(k_x,k_z,\omega )$ in the $k_x$–$k_z$ plane for specific frequencies (a) $\omega h/u_\tau =245$ and (b) $\omega h/u_\tau =716$. Panel (c) shows a zoom-in view of the dash–dotted box region around $(k_x,k_z)=(0,0)$ in (a). The convective peaks are marked by the cross-symbols. In (c), the black dashed box denotes the area in which the isopleths form a valley, and the acoustic peak is located in the red circle. Note that in (a,b), the isopleth level ranges from $10^{-13}$ to $10^{-6}$, while in (c), it ranges from $10^{-11}$ to $10^{-5}$. The Reynolds number is $Re_\tau = 998$.

5.1. Mathematical description of the decomposition

According to (1.1), the pressure fluctuations can be decomposed into two components as

(5.1)\begin{equation} p = p_r + p_s. \end{equation}

The rapid pressure fluctuations $p_r$ and slow pressure fluctuations $p_s$ are governed by the following Poisson equations:

(5.2)\begin{equation} \left. \begin{array}{l@{}} \displaystyle {{\nabla ^2}{p_r} = {f_r} ={-} 2\rho\dfrac{{\partial {u_i}}}{{\partial {x_j}}} \dfrac{{\partial {U_j}}}{{\partial {x_i}}}}, \\ \displaystyle {{\nabla ^2}{p_s} = {f_s} ={-} \rho\dfrac{{{\partial ^2}}}{{\partial {x_i}\partial {x_j}}}({u_i}{u_j} - \overline {{u_i}{u_j}} )}, \end{array} \right\} \end{equation}

respectively, where $f_r$ and $f_s$ denote the corresponding source terms. Because the magnitude of the Stokes pressure induced by the inhomogeneous boundary condition becomes increasingly small as the Reynolds number increases (Gerolymos, Sénéchal & Vallet Reference Gerolymos, Sénéchal and Vallet2013), we adopt the homogeneous Neumann boundary conditions ${{\partial {p_r}/\partial y} |_{y = \pm h}} = {{\partial {p_s}/\partial y} |_{y = \pm h}} = 0$ for both rapid and slow pressure fluctuations following Anantharamu & Mahesh (Reference Anantharamu and Mahesh2020). Performing a Fourier transform over (5.2) results in the following Helmholtz equations:

(5.3)\begin{equation} \left. \begin{array}{l@{}} \displaystyle {\dfrac{{\partial^2 {{\hat p}_r}(\boldsymbol{k},\omega ;y)}}{{\partial {y^2}}} - {k^2}{{\hat p}_r}(\boldsymbol{k},\omega ;y) = {{\hat f}_r}(\boldsymbol{k},\omega ;y)},\quad {\partial {{\hat p}_r}/\partial y{{\rm{|}}_{y ={\pm} h}} = 0}, \\ \displaystyle {\dfrac{{\partial^2 {{\hat p}_s}(\boldsymbol{k},\omega ;y)}}{{\partial {y^2}}} - {k^2}{{\hat p}_s}(\boldsymbol{k},\omega ;y) = {{\hat f}_s}(\boldsymbol{k},\omega ;y)},\quad {\partial {{\hat p}_s}/\partial y{{\rm{|}}_{y ={\pm} h}} = 0}, \end{array} \right\} \end{equation}

where ${\hat p_r}$, ${\hat p_s}$, ${\hat f_r}$ and ${\hat f_s}$ denote the Fourier modes of ${p_r}$, ${p_s}$, ${f_r}$ and ${f_s}$, respectively. The solution of (5.3) can be expressed as an integration of the source term $\hat f_r$ weighted by a Green's function as (Kim Reference Kim1989)

(5.4)\begin{equation} \left. \begin{array}{l@{}} \displaystyle {\hat p_r}(\boldsymbol{k},\omega ;y) = \int_{ - h}^h {G(y,y',k){{\hat f}_r} (\boldsymbol{k},\omega ;y')\,{{\rm d}y}'},\\ \displaystyle {\hat p_s}(\boldsymbol{k},\omega ;y) = \int_{ - h}^h {G(y,y',k){{\hat f}_s} (\boldsymbol{k},\omega ;y')\,{{\rm d}y}'}, \end{array}\right\} \end{equation}

where the Green's function $G(y,y',k)$ is given as

(5.5)\begin{equation} G(y,y',k) = \left\{ {\begin{array}{*{20}{@{}c}} {\dfrac{{\cosh (k(y' - h))\cosh (k(y + h))}}{{2k\sinh (kh)\cosh (kh)}},\quad y \le y'},\\ {\dfrac{{\cosh (k(y' + h))\cosh (k(y - h))}}{{2k\sinh (kh)\cosh (kh)}},\quad y > y'}, \end{array}}\right. \end{equation}

for $k\ne 0$, while for $k=0$ the Green's function is

(5.6)\begin{equation} G(y,y',k) = \left\{ {\begin{array}{*{20}{@{}c}} {\dfrac{1}{2}(y-y'),\quad y \le y'},\\ {\dfrac{1}{2}(y'-y),\quad y > y'}. \end{array}}\right. \end{equation}

In this paper, we focus on the wall pressure fluctuations, and as such the wall-normal coordinate is fixed to $y=\pm h$ without specific declarations.

From (5.1) and (5.3), it is known that the Fourier modes of wall pressure fluctuations can be also decomposed into rapid and slow components as

(5.7)\begin{equation} \hat p(\boldsymbol{k},\omega ) = {\hat p_r}(\boldsymbol{k},\omega ) + {\hat p_s}(\boldsymbol{k},\omega ). \end{equation}

Substituting (5.7) into (2.4) yields the following decomposition of the wavenumber–frequency spectrum of wall pressure fluctuations:

(5.8)\begin{align} {\phi _{pp}}(\boldsymbol{k},\omega ) &= {\phi _{rr}}(\boldsymbol{k},\omega ) + {\phi _{ss}}(\boldsymbol{k},\omega ) + 2{\phi _{rs}}(\boldsymbol{k},\omega ) \nonumber\\ &=\underbrace {\frac{{\overline {{{\hat p}_r}(\boldsymbol{k},\omega ) \hat p_r^*(\boldsymbol{k},\omega )} }}{{{\rm \Delta} {k_x}{\rm \Delta} {k_z}{\rm \Delta} \omega }}}_{{{AS}} - {{Rapid}}} + \underbrace {\frac{{\overline {{{\hat p}_s}(\boldsymbol{k},\omega ) \hat p_s^*(\boldsymbol{k},\omega )} }}{{{\rm \Delta} {k_x}{\rm \Delta} {k_z}{\rm \Delta} \omega }}}_{{{AS}} - {{Slow}}} + \underbrace {2\frac{{{{\rm Re}} \left( {\overline {{{\hat p}_r}(\boldsymbol{k},\omega ) \hat p_s^*(\boldsymbol{k},\omega )} } \right)}}{{{\rm \Delta} {k_x}{\rm \Delta} {k_z} {\rm \Delta} \omega }}}_{{{CS\text{-}RS}}}, \end{align}

where $\phi _{rr}(\boldsymbol {k},\omega )$, $\phi _{ss}(\boldsymbol {k},\omega )$ and $\phi _{rs}(\boldsymbol {k},\omega )$ represent AS-rapid, AS-slow and CS-RS, respectively, and ${\textrm {Re}} ( \cdot )$ denotes the real part of a complex number.

In the early research of wall pressure fluctuations, Kraichnan (Reference Kraichnan1956) proposed a hypothesis that the rapid component of wall pressure fluctuations dominates over the slow component. This hypothesis is supported by the analytical studies of Hodgson (Reference Hodgson1962) and Chase (Reference Chase1980), and the experiments of Johansson, Her & Haritonidis (Reference Johansson, Her and Haritonidis1987). Various predictive models of the spectrum of wall pressure fluctuations are proposed based on Kraichnan's hypothesis (see for example Kraichnan Reference Kraichnan1956; Panton & Linebarger Reference Panton and Linebarger1974; Lee, Blake & Farabee Reference Lee, Blake and Farabee2005; Lysak Reference Lysak2006; Ahn, Graham & Rizzi Reference Ahn, Graham and Rizzi2010). In these models, the wavenumber–frequency spectrum of wall pressure fluctuations $\phi _{pp}(\boldsymbol {k},\omega )$ is approximated as the AS-rapid $\phi _{rr}(\boldsymbol {k},\omega )$. The DNS results of Kim (Reference Kim1989) and Chang III et al. (Reference Chang III, Piomelli and Blake1999) show that the slow component of wall pressure fluctuations has the same order of magnitude as the rapid component. Therefore, it is more accurate to account for both AS-rapid and AS-slow to predict the spectrum of wall pressure fluctuations (see for example Chase Reference Chase1980, Reference Chase1987; Peltier & Hambric Reference Peltier and Hambric2007; Slama et al. Reference Slama, Leblond and Sagaut2018; Grasso et al. Reference Grasso, Jaiswal, Wu, Moreau and Roger2019). In these models, the spectrum of the total wall pressure fluctuations is assumed to be the summation of AS-rapid and AS-slow, while CS-RS is neglected as a consequence of the joint-normal distribution assumption of the velocity fluctuations.

In this section, we use the DNS data to examine the influence of CS-RS on the spectrum of the total wall pressure fluctuations. To conduct quantitative analyses, there are two indicators evaluating the error induced by neglecting CS-RS. The first indicator is the ratio between CS-RS and the spectrum of the total wall pressure fluctuations, viz.

(5.9)\begin{equation} R = \frac{{2{\phi_{rs}}(\boldsymbol{k},\omega )}}{{{\phi _{pp}}(\boldsymbol{k},\omega )}}= \frac{\phi_{pp}(\boldsymbol{k},\omega )-(\phi_{rr}(\boldsymbol{k},\omega )+ \phi_{ss}(\boldsymbol{k},\omega ))}{\phi_{pp}(\boldsymbol{k},\omega )}\times 100\,\%. \end{equation}

When $\phi _{rr}(\boldsymbol {k},\omega )+\phi _{ss}(\boldsymbol {k},\omega )$ is used to approximate the ‘exact value’ of $\phi _{pp}(\boldsymbol {k},\omega )$ as is adopted in many models, the ratio $R$ also represents the relative error induced by neglecting $\phi _{rs}$. It can be proved using the Cauchy–Schwarz inequality that the value of $R$ ranges from $-\infty$ to $50\,\%$. The second indicator is the decibel-scaled error in $\phi _{pp}$ caused by neglecting $\phi _{rs}$, defined as

(5.10)\begin{equation} E = {10 \times {{\log }_{10}}\frac{{{\phi _{rr}}(\boldsymbol{k},\omega ) + {\phi _{ss}}(\boldsymbol{k},\omega )}}{{{\phi _{pp}}(\boldsymbol{k},\omega )}}} = {10 \times {{\log }_{10}}(1-R)}. \end{equation}

Since the spectrum of wall pressure fluctuations is conventionally presented in a decibel scale, we mainly use $E$ as the indicator in this paper. It is worth noting that the magnitudes of $R$ and $E$ are asymmetric for positive and negative values of $\phi _{rs}$. To be specific, if $\phi _{rs}\ge 0$, then $0\,\%\le R\le 50\,\%$ holds, and the corresponding decibel-scaled error $E$ is no smaller than $-3.0$ dB. However, if $\phi _{rs}< 0$, the relative error $R$ can reach $-\infty$ and the corresponding decibel-scaled error $E$ approaches $+\infty$. This suggests that if the value of $\phi _{rs}$ is negative and its magnitude is comparable to $\phi _{rr}+\phi _{ss}$, the wavenumber–frequency spectrum of wall pressure fluctuations can be significantly overestimated.

In the following content of this section, the characteristics of AS-rapid, AS-slow and CS-RS are discussed, and the errors in the spectra of the total wall pressure fluctuations caused by neglecting CS-RS are analysed. In §§ 5.2–5.4, the results of one-dimensional, two-dimensional and three-dimensional spectra at ${Re}_{\tau} =998$ are presented. The effects of Reynolds number are examined in § 5.5.

5.2. Decomposition of one-dimensional spectra

Figure 8 compares the magnitudes of one-dimensional AS-rapid, AS-slow and CS-RS. Note that $\phi _{rs}$ can be negative valued, such that its absolute value $|\phi _{rs}|$ is used to represent its magnitude in figure 8. To validate our results, the spectra of Abe et al. (Reference Abe, Matsuo and Kawanura2005) for AS-rapid and AS-slow with respect to the streamwise wavenumber $k_x$ are superposed in figure 8(a). It is seen that the present results agree with Abe et al. (Reference Abe, Matsuo and Kawanura2005).

Figure 8.

One-dimensional AS-rapid (red solid lines), AS-slow (blue dashed lines) and the absolute value of CS-RS (green dash–dotted lines) with respect to the (a) streamwise wavenumber $k_x$, (b) spanwise wavenumber $k_z$ and (c) frequency $\omega$. The Reynolds number is $Re_\tau = 998$. The AS-rapid (red diamonds) and AS-slow (blue diamonds) with respect to the streamwise wavenumber $k_x$ of Abe et al. (Reference Abe, Matsuo and Kawanura2005) are superposed in panel (a) for comparison.

For the $k_x$- and $\omega$-spectra (see figures 8a and 8c, respectively), the magnitudes of $\phi _{rr}$ and $\phi _{ss}$ are comparable throughout the entire resolved range of $k_x$ and $\omega$, except that the magnitude of $\phi _{ss}$ is slightly larger than that of $\phi _{rr}$ at small $k_x$ or $\omega$. For the $k_z$-spectra, $\phi _{rr}$ is larger than $\phi _{ss}$ at small wavenumbers for $k_zh<10$, but becomes smaller than $\phi _{ss}$ at large wavenumbers for $k_zh>100$. These observations are consistent with the results of Kim (Reference Kim1989) and Chang III et al. (Reference Chang III, Piomelli and Blake1999) at ${Re}_{\tau} =180$. The magnitudes of $\phi _{rr}$ and $\phi _{ss}$ are much larger than those of $|\phi _{rs}|$ at low wavenumbers ($k_xh<50$, $k_zh<50$) and low frequencies ($\omega h/u_\tau <500$), indicating that neglecting CS-RS causes little error in the one-dimensional spectra of the wall pressure fluctuation at these wavenumbers and frequencies. As the wavenumber or frequency increases, the magnitudes of $|\phi _{rs}|$, $\phi _{rr}$ and $\phi _{ss}$ become comparable at high wavenumbers ($k_xh>100$, $k_zh>200$) or high frequencies ($\omega h/u_\tau >1000$).

Figure 9 shows the decibel-scaled errors in the one-dimensional spectra of the total wall pressure fluctuations caused by neglecting CS-RS. At small wavenumbers or frequencies, $E$ is positive valued, and smaller than 1.0 dB. As the wavenumber or frequency increases, the value of $E$ decreases and crosses the dashed line denoting $E=0$ at ${k_x}h \approx 3.5$, ${k_z}h \approx 5.0$ and $\omega h/{u_\tau } \approx 80$. As the wavenumber or frequency continues to increase, the value of $E$ further decreases to its minimum and then increases slightly at very high wavenumbers or frequencies. The minimum values of $E$ for the $k_x$-, $k_z$- and $\omega$-spectra are approximately $-$2.1, $-$1.1 and $-$1.8 dB, respectively, suggesting that the assumption of neglecting CS-RS is acceptable for the one-dimensional spectra of wall pressure fluctuations.

Figure 9.

Decibel-scaled errors in the one-dimensional (a) $k_x$-, (b) $k_z$- and (c) $\omega$-spectra of the wall pressure fluctuation caused by neglecting CS-RS.

5.3. Decomposition of the two-dimensional spectrum

In this subsection, we continue to investigate the decomposition of the two-dimensional spectrum of wall pressure fluctuations. To keep the paper concise, we follow Slama et al. (Reference Slama, Leblond and Sagaut2018) to focus on the $k_x$–$\omega$ spectrum, since the energy-containing convective peak, which is important for the cabin noise generation of a high-speed subsonic civil aircraft (Graham Reference Graham1997), can be identified from the $k_x$–$\omega$ spectrum.

The isopleths of two-dimensional AS-rapid and AS-slow in the $k_x$–$\omega$ plane are shown in figures 10(a) and 10(b), respectively. The difference between $\phi _{rr}$ and $\phi _{ss}$ mainly lies in the low-wavenumber region. As shown in figure 10(a), the isopleths of $\phi _{rr}$ contract to $({k_x},\omega ) = (0,0)$ as $k_x$ approaches zero. This property of $\phi _{rr}$ can be explained using the Helmholtz equation (5.3) of the Fourier modes of the rapid pressure fluctuations.

Figure 10.

Contours of two-dimensional $k_x$–$\omega$ spectra of the (a) rapid and (b) slow components of the wall pressure fluctuation. Six equal-ratio isopleths from $1\times 10^{-5}$ to $1\times 10^{-10}$ are shown by the dashed lines. The convection lines $k_x=\omega /U_c$ are demarcated using the dash–dotted lines. The Reynolds number is $Re_\tau = 998$.

The source term of the rapid pressure fluctuations in the wavenumber–frequency space, i.e. ${\hat f}_r(\boldsymbol {k},\omega ;y')$ in (5.4), can be written as

(5.11)\begin{align} {\hat f_r}(\boldsymbol{k},\omega ;y) &= \frac{1}{{{{(2{\rm \pi} )}^3}}} \int {{\rm d}x} \int {{\rm d} z} \int {{\rm d} t } \cdot \left( { - 2\frac{{{\rm d} U}}{{{\rm d}y}} \frac{{\partial v}}{{\partial x}}} \right)\exp [ - {\rm i}({k_x}x + {k_z}z - \omega t)] \nonumber\\ &={-} \frac{{2{\rm{i}}{k_x}}}{{{{(2{\rm \pi} )}^3}}}\frac{{{\rm d} U(y)}}{{{\rm d}y}} \hat v(\boldsymbol{k},\omega ;y), \end{align}

from which it is understood that ${\hat f}_r(k_x=0,k_z,\omega ;y)=0$. Furthermore, (5.5) and (5.6) show that the Green's function approaches a non-zero finite value as $k_x\to 0$, and, therefore, ${\hat p_r}(k_x=0,k_z,\omega ;y)=0$ holds. Integrating $\phi _{rr}(k_x,k_z,\omega )$ over the spanwise wavenumber $k_z$ yields the same constraint for the two-dimensional spectrum $\phi _{rr}(k_x,\omega )$. This constraint also leads to the contracting behaviour of the isopleths of $\phi _{pp}$ around $k_x=0$ as shown in figure 3. In contrast, the source term of the slow pressure fluctuations does not have such a constraint at $k_x=0$. Consequently, the isopleths of AS-slow do not show a contracting feature around $k_x=0$ (figure 10b).

Both $\phi _{rr}$ and $\phi _{ss}$ show a convective property. Specifically, the large spectral values are concentrated around the convection lines demarcated by the dash–dotted lines. The slopes of the convection lines are identical to the bulk convection velocity, which can be determined using (3.1). The values of the bulk convection velocity for $\phi _{rr}$ and $\phi _{ss}$ are $11.5{u_\tau }$ and $11.7{u_\tau }$, respectively, close to the bulk convection velocity of the total wall pressure fluctuations, i.e. $11.6{u_\tau }$ as shown in figure 3(a). Figure 11 further compares the wavenumber-dependent convection velocities of the rapid, slow and total wall pressure fluctuations. Here, the wavenumber-dependent convection velocity of an arbitrary variable is defined using its wavenumber–frequency spectrum $\phi (k_x,\omega )$ as (del Álamo & Jiménez Reference del Álamo and Jiménez2009)

(5.12)\begin{equation} {U_{c}}({k_x}) = \frac{{\int {\omega \phi ({k_x},\omega )\,{\rm d}\omega } }}{{{k_x} \int {\phi ({k_x},\omega )\,{\rm d}\omega } }}. \end{equation}

From figure 11, it is seen that the values of ${U_{c}}({k_x})$ for the rapid, slow and total wall pressure fluctuations all decrease monotonically as $k_x$ increases. At low wavenumbers for $k_xh<10$, the convection velocity of the rapid wall pressure fluctuations is larger than that of the slow wall pressure fluctuations. For $k_xh>10$, the convection velocities of the rapid, slow and total wall pressure fluctuations are close to each other. They all approach $10.0u_\tau$ as $k_x$ increases. This value is close to the convection velocity of the total wall pressure fluctuations at ${{Re}}_{\tau }=180$ (Choi & Moin Reference Choi and Moin1990). These results indicate that the convective properties of the rapid and slow pressure are similar, and the same convection velocity can be used to model both AS-rapid and AS-slow (Chase Reference Chase1980, Reference Chase1987).

Figure 11.

Wavenumber-dependent convection velocities of the rapid, slow and total wall pressure fluctuations. The Reynolds number is $Re_\tau = 998$.

Figure 12 shows the contours of CS-RS in the $k_x$–$\omega$ plane with the positive and negative values coloured by red and blue, respectively. It is seen that CS-RS is positive in most regions around the convection line (denoted by the dash–dotted line), except for a small region below the convection line at relatively low $k_x$. Besides, since CS-RS also satisfies the constraint $\phi _{rs}(k_x,\omega )\to 0$ as $k_x$ approaches zero, the isopleths show similar contracting behaviour as $\phi _{rr}(k_x,\omega )$ (see figure 10a) at low streamwise wavenumbers.

Figure 12.

Contours of two-dimensional CS-RS in the $k_x$–$\omega$ plane. The positive and negative values are coloured by red and blue, respectively. The convection line $k_x=\omega /U_c$ is demarcated using the dash–dotted line. The Reynolds number is $Re_\tau = 998$.

Figure 13 shows the contours of the decibel-scaled error $E$ in the $k_x$–$\omega$ spectrum of wall pressure fluctuations caused by neglecting CS-RS. The positively and negatively valued regions are coloured by red and blue, respectively. It is seen that there are three local peaks, two in the blue region with negative value of $E$ (marked by the white cross symbols), and one in the red region with positive value of $E$ (marked by the black cross symbol). The details of these peaks, including their locations and values, are given in the figure. Specifically, there are two negative peaks, with one located in the subconvective region and the other in the viscous region. The values of $E$ for these two peaks are $-$2.0 and $-$2.5 dB, respectively. The positive peak is located below the convection line, and its decibel-scaled value is 1.4 dB.

Figure 13.

Contours of the decibel-scaled error $E$ in the $k_x$–$\omega$ spectrum of wall pressure fluctuations caused by neglecting CS-RS. The local peaks in the regions with negative and positive values of $E$ are marked using the white and black cross symbols, respectively. The Reynolds number is $Re_\tau = 998$.

In summary, neglecting CS-RS leads to a maximum decibel-scaled error of approximately $-$2.5 dB at viscous scales for the $k_x$–$\omega$ spectrum of wall pressure fluctuations. This is still an acceptable error magnitude, though it is slightly larger than that in the one-dimensional spectra. In § 5.4, we continue to show that for the three-dimensional spectrum of wall pressure fluctuations, a much larger decibel-scaled error of approximately 4.7 dB in the subconvective region is induced by neglecting CS-RS.

5.4. Decomposition of the three-dimensional spectrum

Figure 14 compares the variations of $\phi _{rr}$, $\phi _{ss}$ and $|\phi _{rs}|$ with respect to $k_x$ at $k_z=0$ and two given frequencies $\omega h/{u_\tau } = 245$ and $\omega h/{u_\tau } = 716$. The corresponding convective wavenumbers demarcated by the vertical dashed lines are $k_xh=15$ and $k_xh=48$, respectively. From these convective wavenumbers and frequencies, it can be calculated that the convection velocities at these two frequencies are $16.3u_{\tau }$ and $14.9u_{\tau }$, respectively. These values are larger than the convection velocity shown in figure 11. Such a difference is attributed to the fact that figure 14 shows the three-dimensional spectrum at a specific spanwise wavenumber $k_z=0$, while in figure 11, the convection velocity of the two-dimensional spectrum averaged over all spanwise wavenumbers is depicted. The above comparison indicates that the convection velocity also depends on the spanwise wavenumber. This is consistent with the results of del Álamo & Jiménez (Reference del Álamo and Jiménez2009).

Figure 14.

Absolute values of AS-rapid $\phi _{rr}$, AS-slow $\phi _{ss}$ and CS-RS $|\phi _{rs}|$ for the three-dimensional wavenumber–frequency spectra of wall pressure fluctuations for $k_z=0$ and two given frequencies (a) $\omega h/{u_\tau } = 245$ and (b) $\omega h/{u_\tau } = 716$. The convective wavenumbers are demarcated using vertical dashed lines. The Reynolds number is $Re_\tau = 998$.

The spectra at the two frequencies have some common features. First, in the subconvective region (left-hand side of the vertical dashed lines), the magnitudes of $|\phi _{rs}|$ and $\phi _{rr}$ are comparable, while that of $\phi _{ss}$ is large. Second, as $k_x$ increases to the convective wavenumber, the magnitudes of $\phi _{rr}$ and $\phi _{ss}$ become close to each other gradually, while $|\phi _{rs}|$ becomes one order of magnitude smaller than $\phi _{rr}$ and $\phi _{ss}$ (shown by the horizontal dotted lines in figure 14). Finally, the magnitudes of $|\phi _{rs}|$, $\phi _{rr}$ and $\phi _{ss}$ become close to each other for $k_xh>200$.

Next, we focus on the error in the three-dimensional spectrum of wall pressure fluctuations caused by neglecting CS-RS. By examining the decibel-scaled error in the $k_x$–$k_z$ plane at various frequencies (not shown for brevity), we find that for an arbitrarily given combination of $(k_x, \omega )$, the peak of $E$ always occurs at $k_z = 0$. Therefore, we focus on $k_z = 0$ in the following error analyses. Figure 15 shows the $k_x$-variation of the decibel-scaled error $E$ at $k_z=0$ and two frequencies $\omega h/{u_\tau } = 245$ and $\omega h/{u_\tau } = 716$. Since the raw data (denoted by the grey lines in figure 15) fluctuate violently, we follow Baars, Hutchins & Marusic (Reference Baars, Hutchins and Marusic2016) to adopt a moving average filter to show the qualitative trend. The moving average filter is expressed as

(5.13)\begin{equation} {E_{{{filt}}}}({k_x}) = \sum_{i ={-} l}^l {\frac{1}{{2l + 1}}E({k_x} + i{\rm \Delta} {k_x})}, \end{equation}

where the filter width is chosen to be $l=2$. The filtered results are denoted by the black lines in figure 15. For both frequencies, the maximum values of $E$ (denoted by red circles in figure 15) occur in the subconvective region. At the lower frequency $\omega h/u_\tau =245$, the maximum values of unfiltered and filtered error reach 5.1 and 4.1 dB, respectively. At the higher frequency, these two values decrease to 2.9 and 2.5 dB, respectively. The results shown in figure 15 indicate a significant overprediction of the spectral magnitude in the subconvective region.

Figure 15.

Unfiltered (grey lines) and filtered (black lines) error $E$ in the three-dimensional wavenumber–frequency spectrum caused by neglecting CS-RS for $k_z=0$ and two given frequencies (a) $\omega h/{u_\tau } = 245$ and (b) $\omega h/{u_\tau } = 716$. The convective wavenumbers are demarcated using vertical dashed lines. The Reynolds number is $Re_\tau = 998$.

While figure 15 shows the error at specific frequencies, it is also useful to find the maximum error over all frequencies. For this purpose, we define the frequency-dependent maximum error ${E_{max}}(\omega )$ as

(5.14)\begin{equation} {E_{max}}(\omega ) = \max_{{k_x}} E({k_x},{k_z} = 0,\omega). \end{equation}

Figure 16 shows the variation of $E_{max}$ with respect to the dimensionless frequency $\omega h / u_\tau$. The raw data and filtered data are denoted by the grey and red lines, respectively. The peak of $E_{\max }$ occurs at a low frequency, of which the value is 4.7 and 5.2 dB with and without the application of the filter, respectively. Such a decibel-scaled error (4.7 dB) corresponds to a large relative error of approximately 195 % caused by neglecting CS-RS in the model of the three-dimensional spectrum of wall pressure fluctuations.

Figure 16.

Unfiltered (grey line) and filtered (red line) frequency-dependent maximum error $E_{max}$ in the three-dimensional wavenumber–frequency spectrum of wall pressure fluctuations caused by neglecting CS-RS. The Reynolds number is $Re_\tau = 998$.

It should be noted here that in practical applications, the total wall pressure fluctuations are a weighted integration of their spectrum at all wavenumbers and frequencies. Therefore, the error in the subconvective region can be attenuated. This is supported by the fact that the errors in one- and two-dimensional spectra are smaller than those in the three-dimensional spectrum. However, the findings of this section cast a doubt on the conventional assumption that CS-RS is negligible compared with AS-rapid and AS-slow in some specific applications. In particular, neglecting CS-RS induces significant errors in the subconvective region at low frequencies. As pointed out by Graham (Reference Graham1997), the properties of the wavenumber–frequency spectrum model in the subconvective region are crucial for the prediction of noise generation in low-Mach-number flows over stiff structures. The omission of CS-RS can be, therefore, inaccurate in corresponding applications, such as automobiles and submarine vehicles, and a modification is expected.

5.5. Reynolds number effects

The above analyses of the error in the spectrum of wall pressure fluctuations caused by neglecting CS-RS are conducted at ${{Re}}=998$. In this subsection, we further examine the effect of Reynolds number. Because the errors in the one- and two-dimensional spectra are acceptable, while those in the three-dimensional spectrum are significantly larger, we focus on the three-dimensional spectrum in this subsection to keep the paper concise.

Figure 17 compares the variations of the filtered value of $E_{max}$ (see (5.14) for definition) with respect to the viscous-scaled frequency for various Reynolds numbers. It is observed that the distributions of $E_{max}$ are close to each other for $330\lesssim {Re}_{\tau} \lesssim 1000$ under the viscous scaling. Specifically, the peak occurs at $\omega ^+\approx 0.25$ and the peak value is approximately 4.7 dB. For the lowest Reynolds number ${Re}_{\tau} =180$, the peak is located at a slightly lower frequency $\omega ^+\approx 1.5$ and the peak value is approximately 6.0 dB. The difference might be induced by the low-Reynolds-number effect. This observation indicates that the maximum error in the wavenumber–frequency spectrum of wall pressure fluctuations induced by neglecting CS-RS is correlated to the property of the velocity field in the near-wall region.

Figure 17.

Variation of filtered $E_{max}$ with respect to the viscous-scaled frequency $\omega ^+$ for different Reynolds numbers.