8.4 Descriptions of turbulence for aeroacoustic analysis

8.4.1 Time correlations and frequency spectra of a single variable

Consider the situation shown in Fig. 8.2 where noise is being radiated by the continuous passage of turbulence over the leading edge of an airfoil. Both, the time variations of the flow properties used to describe the turbulent source (whether they be velocity or pressure) and the sound waves that are heard by a far field observer are stochastic and time stationary. It is important that we have a quantitative measure of the typical frequency content of these signals because we need to assess the impact of the sound on a human listener, and because we are interested in separating out the contributions of different turbulence scales to the acoustic source. In Chapter 3 we introduced the Fourier transform as a way to extract the frequency components of a specific waveform. Applied to a generic flow or acoustic quantity a(t) that varies with time and has zero mean, this definition is

$\tilde{a}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle {\int }_{-T}^{T}a\left(t\right)e^{i\omega t}dt}$

The Fourier transform $\tilde{a}\left(\omega \right)$ is not by itself a very useful measure since, just like a(t), it will vary stochastically. The appropriate average measure of the frequency content is given by the autospectral density of a (often referred to as the autospectrum, the power spectrum, or just the spectrum) defined as

$S_{\mathit{aa}}\left(\omega \right)\equiv \frac{1}{2\pi }{\displaystyle {\int }_{-T}^T{R}_{\mathit{aa}}\left(\tau \right)e^{i\omega \tau }d\tau }$

where R_aa(τ) is the time delay autocorrelation function, defined as

$R_{\mathit{aa}}\left(\tau \right)=E\left[a\left(t\right)a\left(t+\tau \right)\right]$

As can be seen from this expression R_aa(τ) is the average of the signal multiplied by itself at a later time. For a time stationary signal the expected value of a(t)a(t+τ) will not depend on t. The inverse Fourier transform relates the spectrum back to R_aa(τ).

$R_{\mathit{aa}}\left(\tau \right)={\displaystyle {\int }_{-\infty }^{\infty }{S}_{\mathit{aa}}\left(\omega \right)e^{-i\omega \tau }d\omega }$

The definition (8.4.3) tells us that the autocorrelation function is even, i.e., it is symmetric about $\tau =0$, because

$E\left[a\left(t\right)a\left(t+\tau \right)\right]=E\left[a\left(t-\tau \right)a\left(t\right)\right]=E\left[a\left(t\right)a\left(t-\tau \right)\right]$

This means that the spectrum is a real and even function of frequency because Eq. (8.4.2) can be expanded as

$S_{\mathit{aa}}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle {\int }_{-T}^T{R}_{\mathit{aa}}\left(\tau \right)cos\left(\omega \tau \right)d\tau }+\frac{i}{2\pi }{\displaystyle {\int }_{-T}^T{R}_{\mathit{aa}}\left(\tau \right)sin\left(\omega \tau \right)d\tau }$

The imaginary term is zero because R_aa(τ) is even and sin(ωτ) is an odd function. It also follows that S_aa(ω) is an even function of frequency because cos(ωτ) is an even function of frequency.

To illustrate why the concepts of the spectrum and correlation are useful physically, consider the example shown in Fig. 8.3. Fig. 8.3A shows part of a velocity signal u₁(t) measured in boundary layer turbulence near a wall where the mean flow velocity is close to 20 m/s. The signal appears quite random but, as we will see, does have a definite statistical character that reveals important information about the boundary layer turbulence. In Fig. 8.3B the time delay correlation of this signal is shown. The expected value required in Eq. (8.4.3) was obtained by calculating the product $u_1\left(t\right)u_1\left(t+\tau \right)$ for every time instant at which the signal was measured (about 400,000 times in this particular case) and then taking the mean value of those numbers.

The correlation function measures how similar the signal is to its time shifted copy. At zero time delay the signals are a perfect match and the correlation has its maximum value equal, by definition, to the velocity variance $\overline{u_1^2}$ (i.e., its mean square). For nonzero time delay, the correlation decays as the time shifted signal copy becomes less and less similar to the original, reaching 8% of $\overline{u_1^2}$ at about 0.02 s. The overall width of the correlation peak is a measure of the time scale of the largest turbulence and, indeed, the larger scales in the time signal of Fig. 8.3A do appear to be about 0.02 s. To precisely quantify this we introduce the concept of the integral timescale $\mathcal{T}$, defined as

$\mathcal{T}\equiv \frac{1}{E\left[a^2\left(t\right)\right]}{\displaystyle {\int }_0^{\infty }{R}_{\mathit{aa}}\left(\tau \right)d\tau }={\displaystyle {\int }_0^{\infty }{\rho }_{\mathit{aa}}\left(\tau \right)d\tau }$

We also use this opportunity to introduce the correlation coefficient function ρ_aa, which is just R_aa normalized on the variance, so that ${\rho }_{\mathit{aa}}\left(0\right)=1$. In this example ${\rho }_{u_1{u}_1}\left(\tau \right)$ integrates to a time scale of 0.0064 s. The integral time scale is usually about one third of the overall half-width of the correlation peak. If we assume that all the turbulence is traveling with a mean speed of 20 m/s then this timescale can be used to estimate the streamwise lengthscale of the flow (0.0064×20=0.128 m). In general, this type of time-to-space conversion is referred to as Taylor's frozen flow hypothesis. While it is commonly used, and probably reasonably accurate in this example, it is important to remember that it unrealistically treats the turbulence as though it didn't have velocities itself. It can therefore be misleading, particularly in flows where the turbulent velocity fluctuations are significant compared to the mean velocity.

It is clear that the correlation function also has information about smaller turbulence scales. For example, its initial rate of decay will reflect how much of the mean square velocity fluctuation is due to the smallest turbulence. Taking the Fourier transform to obtain the spectrum reveals this information in a more explicit way. Fig. 8.3C shows the spectrum of the velocity signal $S_{u_1{u}_1}\left(\omega \right)$ calculated according to Eq. (8.4.2). Both axes of the spectrum have been plotted on logarithmic scales to more clearly reveal the behavior in different frequency ranges. We see that the overall form of the spectrum is broken into three parts. At low frequencies, where it represents the largest motions of the boundary layer, the spectrum curves downward from a plateau. The level of this plateau is characterized by the zero frequency value of the spectrum (indicated by the horizontal line in Fig. 8.3C) which, in this case gives $S_{u_1{u}_1}\left(0\right)=0.181{\mathrm{m}}^2/{\mathrm{s}}^2/\left(\mathrm{rad}/\mathrm{s}\right)$. Looking at Eq. (8.4.2) for zero frequency, we see that this value corresponds to the integral time scale, scaled as $\mathcal{T}\overline{u_1^2}/\pi$ or, in general $S_{\mathit{aa}}\left(0\right)=\mathcal{T}\overline{a^2}/\pi$. In the mid-frequency range we see that our example velocity spectrum becomes almost straight with a slope on the log-log scale of close to −5/3. As will be discussed in Chapter 9, this is indicative of an inertial subrange behavior in the mid-range scales of the boundary layer turbulence, as hypothesized by Kolmogorov. At the highest frequencies, which are assumed to be generated by the smallest eddies, the spectrum curves downward away from the −5/3^rds slope as a result of the dissipation of these eddies by viscous action.

In general, the spectrum of a time history a(t) can be physically interpreted as revealing the contributions to the mean square fluctuation $\overline{a^2}$ at each frequency. This can be demonstrated very simply using the inverse transform relationship, Eq. (8.4.4) which, for zero time delay τ becomes

$R_{\mathit{aa}}\left(0\right)=\overline{a^2}={\displaystyle {\int }_{-\infty }^{\infty }{S}_{\mathit{aa}}\left(\omega \right)d\omega }$

This is known as Parseval's theorem which states that the “power” in the time and frequency domains is the same, and that the spectrum divides up that power by frequency. (Note that the term “power” is loosely used in spectral analysis to refer to the mean square.) It is in this sense that S_aa(ω) is a spectral density function. We also see that the units of S_aa(ω) will be the units of $\overline{a^2}$ per radian-per-second.

One simple but critical detail here is that the mathematical definition of the spectral density includes both positive and negative frequencies. In the present context of one-dimensional spectra these mean the same thing. However, note from Eq. (8.4.6) that the energy is spread over both the positive and negative domains. We call the spectrum double sided and this is the norm for mathematical analysis. However, in many situations including presenting predictions or measurements of sound spectra, it is normal to consider them to exist only for positive frequencies and to double the spectral values. We then refer to the spectrum as single sided for which we introduce the symbol G_aa(ω).

The fact that S_aa(ω) (and by extension G_aa(ω)) is a density function means that we expect it to integrate to a defined physical quantity. This limits the ways in which we may scale it and the frequency variable upon which it depends. For example, if we wanted to express our spectrum as a function of frequency f in cycles per second or Hertz, then we would write S_aa(f)=2πS_aa(ω) since we are expecting that

$\overline{a^2}={\displaystyle {\int }_{-\infty }^{\infty }{S}_{\mathit{aa}}\left(f\right)df}$

and df=dω/2π. Another way of thinking about this is to envision S_aa(f) as resulting in a Fourier transform defined in terms of frequency in Hertz as,

$S_{\mathit{aa}}\left(f\right)={\int }_{-T}^T{R}_{\mathit{aa}}\left(\tau \right)e^{2\mathit{\pi if\tau }}d\tau$

so comparing this with Eq. (8.4.2) we see that S_aa(f)=2πS_aa(ω). A more complicated example is the scaled velocity spectrum, which we might reasonably want to normalize so that it integrates to $\overline{u_1^2}/U_{\infty }^2$ where $U_{\infty }$ is the free stream velocity. The integrand of Eq. (8.4.7) is then $S_{u_1{u}_1}\left(\omega \right)/U_{\infty }^2$. However, at the same time it would be physically meaningful to nondimensionalize the frequency as $\sigma =\omega L/U_{\infty }$ where L is a representative physical scale of the flow. To preserve the integral under the spectrum our spectral normalization must become $S_{u_1{u}_1}\left(\sigma \right)=S_{u_1{u}_1}\left(\omega \right)U_{\infty }/L$ so that

$\overline{\frac{u_1^2}{U_{\infty }^2}}={\int }_{-\infty }^{\infty }\frac{S_{u_1{u}_1}\left(\sigma \right)}{U_{\infty }^2}d\sigma \sigma =\omega L/U_{\infty }$

Some quite specific conventions exist for presenting broadband sound spectra. We present acoustic pressure spectra in decibels as using the “narrow band” sound pressure level (SPL), which for a spectrum is defined as

$\mathrm{S}\mathrm{P}\mathrm{L}=10{log}_{10}\left[G_{\mathit{pp}}\left(\sigma \right)/\left(p_{ref}^2/\mathrm{\Delta }{\sigma }_{ref}\right)\right]\mathrm{dB}\mathrm{r}\mathrm{e}{p}_{ref}/\mathrm{\Delta }{\sigma }_{ref}^{1/2}$

where △σ_ref refers to whatever frequency unit is being used for σ (e.g., Hz, radians per second, or U_∞/L) and p_ref is the reference pressure, conventionally taken as 20 μPa for measurements in air. Note that the single sided spectrum G_pp is used to express the SPL. We refer to Eq. (8.4.8) as “narrow band” to distinguish it from one-third octave band SPL. As discussed in Chapter 1, the third octave spectrum divides the spectrum into frequency bands, with each band being 2^1/3 times the size of the preceding band. In this case the SPL in the nth band is given by

${\mathrm{S}\mathrm{P}\mathrm{L}}_n=10{log}_{10}\left(\frac{1}{p_{\mathrm{r}\mathrm{e}\mathrm{f}}^2}{\displaystyle {\int }_{f_l^{\left(n\right)}}^{f_u^{\left(n\right)}}{G}_{\mathit{pp}}\left(f\right)df}\right)\mathrm{dB}\mathrm{r}\mathrm{e}{p}_{\mathrm{r}\mathrm{e}\mathrm{f}}$

where the lower and upper frequency limits of each band are defined in terms of the mid-band frequency f_n as

$\begin{array}{l}{f}_l^{\left(n\right)}=f_n/2^{1/6}\\ f_u^{\left(n\right)}=f_n\times 2^{1/6}\end{array}$

The mid-band frequencies are calculated using Eq. (1.3.3). The third octave bands appear evenly spaced when plotted on a logarithmic scale, and the spectral levels represented are no longer densities but pure mean square contributions, the frequency dependence having been integrated out in Eq. (8.4.9).

To close this section, we return to the definition of the spectrum S_aa(ω) given in Eq. (8.4.2) and investigate the relationship between this average measure of the frequency content, and the Fourier transform of the instantaneous signal ã(ω). Since we are restricting ourselves to a time stationary signal we can average the time delay correlation (Eq. 8.4.3) over time without changing its value, so

$R_{\mathit{aa}}\left(\tau \right)=\frac{1}{2T}{\displaystyle {\int }_{-T}^{T}E\left[a\left(t\right)a\left(t+\tau \right)\right]dt}$

Substituting this into Eq. (8.4.2) we obtain

$S_{\mathit{aa}}\left(\omega \right)=\frac{1}{2\pi }{\int }_{-T}^T\frac{1}{2T}{\int }_{-T}^{T}E\left[a\left(t\right)a\left(t+\tau \right)\right]dt{e}^{i\omega \tau }d\tau =\frac{1}{4\pi T}E\left[{\int }_{-T}^T{\int }_{-T}^{T}a\left(t\right)a\left(t+\tau \right)e^{i\omega \tau }\mathit{d\tau dt}\right]$

Now we make the substitution $t^{\prime }=t+\tau$ and note that $\mathit{dtd\tau }=dtd{t}^{\prime }$ to obtain

$S_{\mathit{aa}}\left(\omega \right)=\frac{1}{4\pi T}E\left[{\displaystyle {\int }_{-T}^T{\displaystyle {\int }_{-T+t}^{T+t}a\left(t\right)a\left(t^{\prime }\right)e^{i\omega \left(t^{\prime }-t\right)}d{t}^{\prime }dt}}\right]$

We next change integration limits from (−T, −T+t) and (T, T+t) to (−T, −T) and (T, T) as illustrated in Fig. 8.4. The expected value of the integrand E[a(t)a(t′)]=R_aa(t−t′), which is only a function of t′−t, and is constant along diagonal lines such as the one shown. The two integration regions are shown by the solid and dashed boxes. We see that the change in limits is only valid if the expected value of the integrand decays to zero in a time t′−t≪T and that the time history itself is bounded by the limits −T<t′<T, as required by the definition of the Fourier transform. With this change, and rearranging Eq. (8.4.13), we obtain

$\begin{array}{c}{S}_{\mathit{aa}}\left(\omega \right)=\frac{1}{4\pi T}E\left[{\displaystyle {\int }_{-T}^{T}a\left(t\right)e^{-i\omega t}}dt{\displaystyle {\int }_{-T}^{T}a\left(t^{\prime }\right)e^{i\omega t^{\prime }}d{t}^{\prime }}\right]\\ =\frac{\pi }{T}E\left[{\tilde{a}}^{⁎}\left(\omega \right)\tilde{a}\left(\omega \right)\right]\end{array}$

where ${\tilde{a}}^{⁎}\left(\omega \right)$ denotes the conjugate of $\tilde{a}\left(\omega \right)$. An important observation here is that the definition of the Fourier transform that is used here does not give the spectrum for zero frequency and is limited to angular frequencies for which ω≫π/T.

Eq. (8.4.14) shows that the spectrum is also the expected value of the magnitude squared of the Fourier transform, supporting the physical interpretation given to it. The equivalence demonstrated in Eq. (8.4.14) is particularly useful when it comes to determining spectra from measured signals—as will be discussed further in Chapter 11. The fact that Eq. (8.4.14) does not apply at $\omega =0$ means that spectra obtained using this method can only be used to estimate the integral scale by assuming that it is revealed by the asymptotic level of the spectrum as ω tends to zero.

The relationship in Eq. (8.4.14) can be cast in a somewhat different form that can be particularly useful in analysis. Specifically, consider the expected value of the right hand side but with two different frequency arguments, that is,

$E\left[{\tilde{a}}^{⁎}\left(\varpi \right)\tilde{a}\left(\omega \right)\right]=\frac{1}{{\left(2\pi \right)}^2}{\displaystyle {\int }_{-T}^T{\displaystyle {\int }_{-T}^{T}E\left[a\left(t\right)a\left(t^{\prime }\right)\right]e^{-i\varpi t}{e}^{i\omega t^{\prime }}dtd{t}^{\prime }}}$

We make the substitution of τ for t′ with τ=t′−t, and as before ignore the change in the limit by assuming large T giving,

$\begin{array}{c}E\left[{\tilde{a}}^{⁎}\left(\varpi \right)\tilde{a}\left(\omega \right)\right]=\frac{1}{{\left(2\pi \right)}^2}{\displaystyle {\int }_{-T}^T{\displaystyle {\int }_{-T}^{T}E\left[a\left(t\right)a\left(t+\tau \right)\right]}}{e}^{i\omega \tau }{e}^{i\left(\omega -\varpi \right)t}\mathit{d\tau dt}\\ =\frac{1}{2\pi }{\displaystyle {\int }_{-T}^T{e}^{i\left(\omega -\varpi \right)t}dt\frac{1}{2\pi }}{\displaystyle {\int }_{-T}^T{R}_{\mathit{aa}}\left(\tau \right)e^{i\omega \tau }d\tau }\end{array}$

The first integral term on the right hand side is equal to δ(ϖ−ω), whereas the second is simply the frequency spectrum. Our final result is therefore,

$E\left[{\tilde{a}}^{⁎}\left(\varpi \right)\tilde{a}\left(\omega \right)\right]=\delta \left(\varpi -\omega \right)S_{\mathit{aa}}\left(\omega \right)$

This shows that for a stationary signal the fluctuations at each frequency are uncorrelated, which can be valuable in the analysis of some turbulent flow phenomena.

8.4.2 Time correlations and frequency spectra of two variables

Consider again the situation illustrated in Fig. 8.2. In addition to characterizing the typical behavior of the turbulent or acoustic variables, we are also interested in characterizing the typical relationships, for example between the sound pressure fluctuations at two points in acoustic the far field from the source. The way to do this is to define, by analogy with Eq. (8.4.3), a time delay cross correlation function

$R_{\mathit{ab}}\left(\tau \right)\text{≡}E\left[a\left(t\right)b\left(t+\tau \right)\right]$

where a and b have zero mean. This measures the similarity of the stochastic variations in a with those in b, as the time delay τ between b and a is varied. We are assuming that both variables are time stationary so that the cross correlation will only be a function of τ. For zero time delay R_ab(0) is equal to the covariance $\overline{\mathit{ab}}$. We define the cross correlation coefficient as

${\rho }_{\mathit{ab}}\left(\tau \right)\text{≡}\frac{R_{\mathit{ab}}\left(\tau \right)}{\sqrt{\overline{a^2}\overline{b^2}}}$

Defined this way $-1\le {\rho }_{\mathit{ab}}\left(\tau \right)\le 1$ and the coefficient only reaches 1 or −1 if a(t) and $b\left(t+\tau \right)$ are exact scaled copies of each other. Unlike the autocorrelation function, R_ab(τ) can be asymmetric and reach its maximum magnitude at a nonzero time delay. In the example of Fig. 8.2 the acoustic signal at A would precede the acoustic signal at B and thus $R_{\mathit{AB}}\left(\tau \right)=E\left[p_A\left(t\right)p_B\left(t+\tau \right)\right]$ would have a positive maximum at positive time delay τ. Inverting the sign of the acoustic signal (say by shifting the point A from below to above the airfoil if the sound is from a lift dipole only) would reverse the sign of R_AB which would then have a negative peak.

Consistent with the autospectrum, we define the cross spectral density as the Fourier transform of the correlation function

$S_{\mathit{ab}}\left(\omega \right)\text{≡}\frac{1}{2\pi }{\displaystyle {\int }_{-T}^T{R}_{\mathit{ab}}\left(\tau \right)e^{i\omega \tau }d\tau }$

Expanding this we obtain

$S_{\mathit{ab}}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle {\int }_{-T}^T{R}_{\mathit{ab}}\left(\tau \right)cos\left(\omega \tau \right)d\tau +\frac{i}{2\pi }}{\displaystyle {\int }_{-T}^T{R}_{\mathit{ab}}\left(\tau \right)sin\left(\omega \tau \right)d\tau }$

showing that S_ab(ω) will, in general, be a complex function because of the probable asymmetry of R_ab(τ). It is straightforward to verify that S_ab will also be conjugate symmetric with frequency, so $S_{\mathit{ab}}\left(\omega \right)=S_{\mathit{ab}}^{⁎}\left(-\omega \right)$. The real and imaginary parts of S_ab are referred to as the cospectrum C_ab and the quad-spectrum Q_ab respectively.

As one would expect, the inverse Fourier transform relates the cross spectrum back to the cross correlation

$R_{\mathit{ab}}\left(\tau \right)={\displaystyle {\int }_{-\infty }^{\infty }{S}_{\mathit{ab}}\left(\omega \right)e^{-i\omega \tau }d\omega }$

For zero time delay we obtain an expression for the covariance in terms of the cross spectral density

$\overline{\mathit{ab}}=R_{\mathit{ab}}\left(0\right)={\displaystyle {\int }_{-\infty }^{\infty }{S}_{\mathit{ab}}\left(\omega \right)d\omega }={\displaystyle {\int }_{-\infty }^{\infty }{C}_{\mathit{ab}}\left(\omega \right)}+i\cdot Q_{\mathit{ab}}\left(\omega \right)d\omega ={\displaystyle {\int }_{-\infty }^{\infty }{C}_{\mathit{ab}}(\omega })d\omega$

The conjugate symmetry of the cross spectrum ensures that quad-spectrum does not contribute to the covariance. The cospectrum may therefore be thought of as the covariance per unit frequency. Two more physically important expressions of the cross spectrum are the coherence and phase spectra, defined respectively as

${\gamma }_{\mathit{ab}}^2\left(\omega \right)=\frac{{\left|S_{\mathit{ab}}\left(\omega \right)\right|}^2}{S_{\mathit{aa}}\left(\omega \right)S_{bb}\left(\omega \right)}$

and

${\theta }_{\mathit{ab}}\left(\omega \right)=arctan\left(\frac{Q_{\mathit{ab}}\left(\omega \right)}{C_{\mathit{ab}}\left(\omega \right)}\right)$

The coherence γ_ab² is a squared and normalized measure of the expected value of the product of the complex amplitude of the two quantities a and b at a given frequency. If, in multiple realizations of our airfoil example, the lift (assumed say to be a compact dipole source) and sound variations (which may include background noise) were randomly phased with respect to each other implying no linear connection, then the coherence between them would be zero. If the phasing were not random, but tended to prefer a particular value, then the coherence would be positive with a value that increases the more closely correlated the signals are at that frequency. For nonzero coherence, the phase spectrum θ_ab(ω) gives the preferred value of the phase difference, and the phase is undefined for ${\gamma }_{\mathit{ab}}^2=0\text{.}$

The coherence is an even function of frequency and is normalized so that $0\le {\gamma }_{\mathit{ab}}^2\le 1$, a value of 1 being achieved when b(t) is a constant multiple of a(t). The phase is an odd function and has a straightforward relationship to the time delay between variables. In the case of the airfoil example we argued that the positive time delay between the sound fluctuations in the far field would lead to R_AB(τ) reaching its maximum magnitude at positive τ. This situation corresponds to a positive gradient of the phase spectrum θ_AB(ω) with frequency.

A review of the above material will make it clear that the auto correlation and auto spectrum functions are simply special cases of the cross correlation and cross spectrum functions with $b\left(t\right)=a\left(t\right)$. This is also true for the relationship between the cross spectral density and the stochastic Fourier transform. Using a derivation entirely analogous to that laid out in Eqs. (8.4.11)–(8.4.14), we can show that, for angular frequencies ω≫π/T,

$S_{\mathit{ab}}\left(\omega \right)=\frac{\pi }{T}E\left[{\tilde{a}}^{⁎}\left(\omega \right)\tilde{b}\left(\omega \right)\right]$

where $\tilde{a}\left(\omega \right)$ and $\tilde{b}\left(\omega \right)$ are the Fourier transforms of a(t) and b(t). Likewise we can show, analogously to the derivation of Eq. (8.4.17) that

$E\left[{\tilde{a}}^{⁎}\left(\varpi \right)\tilde{b}\left(\omega \right)\right]=\delta \left(\varpi -\omega \right)S_{\mathit{ab}}\left(\omega \right)$

Note that if one of the variables involved in the cross spectrum or correlation is a vector then these statistical measures are also vectors, e.g., the pressure velocity spectrum $S_{p{u}_i}$. If both are vectors then the resulting measures are tensors, e.g., the velocity correlation tensor $R_{u_i{u}_j}$, which is commonly abbreviated as R_ij.

8.4.3 Spatial correlation and the wavenumber spectrum

The correlation and spectral analysis techniques we have introduced to analyze the temporal behavior of a turbulent flow can equally well be applied to revealing its spatial structure. Consider turbulent velocity fluctuations u₂ as a function of position along a line x₁ through a turbulent flow (Fig. 8.5). The correlation between simultaneous fluctuations at two points on the line can in general be written as

$R_{22}\left(x_1,x_1^{\prime }\right)=E\left[u_2\left(x_1\right)u_2\left(x_1^{\prime }\right)\right]$

If the flow is assumed homogeneous along the line (i.e., its average properties are independent of x₁), then the correlation will only be a function of the separation between the two points $\mathrm{\Delta }{x}_1=x_1^{\prime }-x_1$

$R_{22}\left(\mathrm{\Delta }{x}_1\right)=E\left[u_2\left(x_1\right)u_2\left(x_1^{\prime }\right)\right]$

Homogeneity also allows us to define an integral lengthscale for the turbulence

$L_{21}\text{≡}\frac{1}{\overline{u_2^2}}{\displaystyle {\int }_0^{\infty }{R}_{22}\left(\mathrm{\Delta }{x}_1\right)d\mathrm{\Delta }{x}_1}$

where L_ij is defined as the integral lengthscale of velocity component u_i in the direction x_j. We can additionally Fourier transform the correlation function in the homogeneous direction to give the wavenumber spectrum

${\varphi }_{22}\left(k_1\right)\text{≡}\frac{1}{2\pi }{\displaystyle {\int }_{-R_{\infty }}^{R_{\infty }}{R}_{22}\left(\mathrm{\Delta }{x}_1\right)e^{-i{k}_1\mathrm{\Delta }{x}_1}d\mathrm{\Delta }{x}_1}$

The inverse relationship giving the correlation function in terms of the wavenumber spectrum is

$R_{22}\left(\mathrm{\Delta }{x}_1\right)={\displaystyle {\int }_{-\infty }^{\infty }{\varphi }_{22}\left(k_1\right)e^{i{k}_1\mathrm{\Delta }{x}_1}d{k}_1}$

Note that, to be consistent with the Fourier transform convention introduced in Chapter 1, the signs of the exponents in the spatial Fourier transform and its inverse are reversed compared to those used with the time transform, Eqs. (8.4.1), (8.4.2). The wavenumber spectrum and spatial correlation function reveal the averaged structure of the turbulence in ways that are entirely analogous to the frequency spectrum and time correlation functions illustrated in Fig. 8.3.

The homogeneity of the turbulent flow is important in the same way that stationarity is important in time histories. Without it, the integral scale and spectrum, as well as properties derived from the spectrum, lose much of their physical meaning and mathematical utility. The assumption of homogeneity will be good if the average properties a flow vary on a scale much larger than the scale of its turbulence. For example, Fig. 8.5 depicts part of a two-dimensional turbulent boundary layer formed under a free stream in the x₁ direction. The largest turbulence scales will be of the order of the boundary layer thickness δ, a distance much smaller than the scale on which the boundary layer is growing. Homogeneity in x₁ is therefore a good assumption in this case. Since the boundary layer is two dimensional it would be a good approximation in the spanwise x₃ direction, but not in direction x₂ perpendicular to the wall.

There is much scope in the correlation and spectrum descriptors that we have not yet acknowledged. It is clear from the boundary layer example that the spatial correlation and wavenumber spectrum will be functions of x₂, could also be taken in the x₃ direction, and could be taken of any velocity component combination. So, we would have L_in(x₂), R_ij(x₂,Δx_n), and ϕ_ij(x₂,k_n), where $n=1,3$. Perhaps more interestingly we can expand correlations and spectra to more than one dimension. Most generally, we could consider the five-dimensional correlation between any two points in the boundary layer which has the form

$R_{ij}\left(\mathrm{\Delta }{x}_1,x_2,x_2^{\prime },\mathrm{\Delta }{x}_3,\tau \right)=E\left[u_i\left(x_1,x_2,x_3,t\right)u_j\left(x_1^{\prime },x_2^{\prime },x_3^{\prime },t^{\prime }\right)\right]$

and we can envision taking the Fourier transform in △x₁ and △x₃ to yield the two-wavenumber spectrum

$\begin{array}{ll}{\varphi }_{ij}\left(x_2,x_2^{\prime },k_1,k_3,\tau \right)=& \frac{1}{{\left(2\pi \right)}^2}{\displaystyle {\int }_{-R_{\infty }}^{R_{\infty }}{\displaystyle {\int }_{-R_{\infty }}^{R_{\infty }}{R}_{ij}\left(\mathrm{\Delta }{x}_1,x_2,x_2^{\prime },\mathrm{\Delta }{x}_3,\tau \right)}}\hfill \\ & {\displaystyle {\displaystyle e^{-i{k}_1\mathrm{\Delta }{x}_1-i{k}_3\mathrm{\Delta }{x}_3}d\mathrm{\Delta }{x}_{1}d\mathrm{\Delta }{x}_3}}\hfill \end{array}$

and then the Fourier transform in time to obtain the wavenumber frequency spectrum

$\begin{array}{ll}{\Phi }_{ij}\left(x_2,x_2^{\text{'}},k_1,k_3,\omega \right)=& \frac{1}{{\left(2\pi \right)}^3}{\int }_{-T}^T{\int }_{-R_{\infty }}^{R_{\infty }}{\int }_{-R_{\infty }}^{R_{\infty }}{R}_{ij}\left(\mathrm{\Delta }{x}_1,x_2,x_2^{\text{'}},\mathrm{\Delta }{x}_3,\tau \right)\\ & e^{i\omega \tau -i{k}_1\mathrm{\Delta }{x}_1-i{k}_3\mathrm{\Delta }{x}_3}d\mathrm{\Delta }{x}_{1}d\mathrm{\Delta }{x}_{3}d\tau \end{array}$

The spectral quantities can of course always be related back to correlations through inverse Fourier transforms. So, for example, the time cross spectrum of the velocities at two points in the flow, (x₁,x₂,x₃) and (x₁′,x₂′,x₃′) can be obtained from the wavenumber frequency spectrum in Eq. (8.4.33) as

$S_{ij}\left(\mathrm{\Delta }{x}_1,x_2,x_2^{\text{'}},\mathrm{\Delta }{x}_3,\omega \right)={\int }_{-\infty }^{-\infty }{\int }_{-\infty }^{\infty }{\Phi }_{ij}\left(x_2,x_2^{\text{'}},k_1,k_3,\omega \right)e^{i{k}_1\mathrm{\Delta }{x}_1+i{k}_3\mathrm{\Delta }{x}_3}d{k}_{1}d{k}_3$

Descriptors like those of Eqs. (8.4.31)–(8.4.33) encompass all the possible second order statistics defined by a flow and are sufficiently rich to represent the complete acoustic source term in many linear aeroacoustic problems. Obviously, in any specific problem, the number of homogeneous directions in the flow will limit the dimensionality of the wavenumber spectrum that can be defined, just as time stationarity of the flow will determine if a physically useful frequency spectrum exists. Note that the two-point correlation, as written in the right hand side of Eq. (8.4.31), can always be defined.

Acoustic sources are often cast in terms of the fluctuating velocity field of the turbulence, but may also appear in terms of the fluctuating vorticity or the pressure on a surface, for example. In our boundary layer example, the fluctuating wall-pressure field will be characterized by the wavenumber frequency spectrum

${\Phi }_{\mathit{pp}}\left(k_1,k_3,\omega \right)=\frac{1}{{\left(2\pi \right)}^3}{\displaystyle {\int }_{-T}^T{\displaystyle {\int }_{-R_{\infty }}^{R_{\infty }}{\displaystyle {\int }_{-R_{\infty }}^{R_{\infty }}{R}_{\mathit{pp}}\left(\mathrm{\Delta }{x}_1,\mathrm{\Delta }{x}_3,\tau \right)e^{i\omega \tau -i{k}_1\mathrm{\Delta }{x}_1-i{k}_3\mathrm{\Delta }{x}_3}d\mathrm{\Delta }{x}_{1}d\mathrm{\Delta }{x}_{3}d\tau }}}$

where

$R_{\mathit{pp}}\left(\mathrm{\Delta }{x}_1,\mathrm{\Delta }{x}_3,\tau \right)=E\left[p\left(x_1,x_3,t\right)p\left(x_1^{\prime },x_3^{\prime },t^{\prime }\right)\right]$

Note that relationships exactly analogous to Eqs. (8.4.14), (8.4.17) exist for the wavenumber domain. Thus, for example, we can write for the wavenumber frequency spectrum of the wall-pressure,

$\frac{\pi }{T}E\left[{\widetilde{\stackrel{~}{p}}}^{⁎}\left(k_1,k_3,\omega \right)\widetilde{\stackrel{~}{p}}\left(k_1^{\text{'}},k_3^{\text{'}},\omega \right)\right]={\Phi }_{\mathit{pp}}\left(k_1,k_3,\omega \right)\delta \left(k_1-k_1^{\text{'}}\right)\delta \left(k_3-k_3^{\text{'}}\right)$

where $\widetilde{\stackrel{~}{p}}$ denotes the wavenumber frequency transform of p′. Also, we can relate the one-dimensional wavenumber spectrum to the one-dimensional wavenumber transform as,

${\varphi }_{\mathit{pp}}\left(k_1\right)=\frac{1}{4\pi R_{\infty }}E\left[{\displaystyle {\int }_{-R_{\infty }}^{R_{\infty }}p\left(\mathrm{\Delta }{x}_1\right)e^{i{k}_1\mathrm{\Delta }{x}_1}dt}{\displaystyle {\int }_{-R_{\infty }}^{R_{\infty }}p\left(\mathrm{\Delta }{x}_1^{\prime }\right)e^{-i{k}_1\mathrm{\Delta }{x}_1^{\prime }}d{t}^{\prime }}\right]=\frac{\pi }{R_{\infty }}E\left[{\widetilde{\stackrel{~}{p}}}^{⁎}\left(k_1\right)\widetilde{\stackrel{~}{p}}\left(k_1\right)\right]$

for example. In addition, by repeated application of Eq. (8.4.38) or (8.4.14) multidimensional relationships are obtained, such as,

${\varphi }_{ij}\left(k_1,k_3\right)=\frac{{\pi }^2}{R_{\infty }^2}E\left[{{\widetilde{\stackrel{~}{u}}}_i}^{⁎}\left(k_1,k_3\right){\widetilde{\stackrel{~}{u}}}_j\left(k_1,k_3\right)\right]$

for the planar wavenumber spectrum of the velocity fluctuations.

The combination of space and time in the correlation function and in the wavenumber frequency spectrum in general captures information about the convection or phase velocity of fluid motions as well as their scale and intensity. We can always calculate a velocity component from Δx_i/τ  or  ω/k_i. Whether such a component is physically meaningful depends on the form of the correlation or spectrum at that point. Where turbulence is carried by a mean flow, much of the energy of the flow will appear concentrated around a convective ridge.

Fig. 8.6 shows this type of feature in the pressure correlation coefficient function $R_{\mathit{pp}}\left(\mathrm{\Delta }{x}_1,0,\tau \right)/\overline{p^2}$ measured underneath a high Reynolds number turbulent boundary layer. The convective ridge defines the elongated form of the correlation function. The convection velocity U_c is calculated as Δx₁/τ at the center of the ridge. For small separations Δx₁, the correlation levels on the ridge are high because the turbulence does not evolve much over short distances. The slope of the convective ridge and U_c are relatively low here in the measurements (about 0.6 U_∞) because the correlation includes a substantial contribution from small turbulence scales in the near-wall region where the flow speeds are slow. At larger separations the correlation values reduce as the turbulence has more distance to evolve. Smaller eddies evolve more quickly and so the correlation increasingly represents the larger eddies of the flow that are more likely to occupy faster flowing regions further from the wall. The slope of the convective ridge and U_c therefore increase for larger separations. The convection velocity of wall-pressure fluctuations in a smooth wall boundary layer is usually observed to be between 60% and 80% of the free stream velocity. The boundary layer wall-pressure spectrum and correlation function will be discussed in more detail in the next chapter.