Conversion of Measured Turbulence Spectra from Temporal to Spatial Domain

Abstract

A novel type of conversion of point-measured temporal turbulence power spectra to wavenumber space is proposed. By converting the temporal measurement records into spatial connected streakline elements, the classical assumption of a local mean velocity in Taylor’s hypothesis can be completely bypassed. The presented method is illustrated with examples from both hot-wire anemometry and laser Doppler velocimetry, but may in principle just as well be applied to any flow field property such as pressure, temperature, concentration, or density. Computer generated data of a large eddy with a sharp modulation frequency as well as a turbulent von Karman spectrum are presented to demonstrate the correctness of the principle. Laser Doppler velocimetry measurements, which in themselves appear to be particularly suitable for application of this technique, taken at different off-center positions in a round turbulent jet are then used to demonstrate the difference between the current and the classical temporal-to-spatial domain conversions. The novel method displays the behavior expected from spatial spectra measured along homogeneous directions in the very same turbulent axisymmetric jet, while the classical Taylor’s hypothesis, as expected, shows increasing deviation further away from the center axis where the turbulence intensity grows rapidly. Interpretation of first-order statistics as well as different kinds of spectral estimates is proposed and discussed.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

The spatial structure of a turbulent velocity field is of great theoretical interest as its kinematics describe the distribution of spatial scales and its dynamics describe their evolution from large energy carrying scales to smaller scales and finally to dissipation.

However, the overwhelming number of turbulence measurements results in time records from stationary probes, either hot-wire probes (hot-wire anemometers, HWA) or laser beam probes (laser Doppler anemometers, LDA). The spatial structure of the turbulent velocity field is then inferred by “Taylor’s hypothesis,” as first presented in [1], assuming a “frozen” velocity field carried past the probe with the local mean velocity. However, Taylor’s hypothesis breaks down at higher turbulence intensities and can then only be applied with additional corrections, see, for example, [2–4].

The spatial structure of the flow field can also be probed by simultaneous measurements at two or more points in space, e.g., by two LDAs or two HWAs. However, these measurements are still subject to distortions due to the varying local velocity. Only by quite advanced techniques such as moving probes has the fluctuating convection velocity issue previously been beaten, but only for computing spatial derivatives of fluctuating velocities [5]. This strategy is, however, not practical for measuring the frequency contents of the flow since the limited record lengths attainable are typically too short to acquire useable spectra.

Whole flow field measurements can supply the instantaneous spatial velocity field and are great for the measurement of large scales. Until now, however, these methods lack the resolution needed for the most interesting small scales near the dissipation range. Furthermore, flow field methods, e.g., particle image velocimetry (PIV) require local homogeneity of the velocity field for meaningful statistical results, while time records merely require stationarity. The power spectrum derived from a temporal record is not wrong, of course. It is just one way of viewing the distribution of turbulence power over frequency as measured by a stationary probe. Power spectra from temporal records show the power of the fluctuations of the velocity of the fluid as it passes the probe. It is different from the energy spectrum of the spatial structures found from a velocity measured as a spatial record of the velocity field at one instance in time, which is basis for fundamental turbulence theory (see, e.g., [6]).

The measurement we propose here is best described as a measurement of the fluid velocity at a chosen point in the flow field, the measurement point (MP) or more realistically in a small measurement volume (MV) surrounding the MP. However, we convert the temporal record of the velocity through the MV to a spatial record of the velocity along the instantaneous streakline through the MP. We do this by measuring simultaneously both the desired velocity component and the velocity vector magnitude. We illustrate the method in Fig. 18.1.

Fig. 18.1

A streakline passing through the measurement point (x ₀, y ₀) in a jet flow at time t. Red: Streakline, s(x, t). MP: position of measurement point, x ₀

The desired velocity component is recorded at the MP as a function of time, either by a time record with constant sampling intervals, as, e.g., in the case of a digitized HWA record, or by a randomly sampled time record as, e.g., in the case of a burst-mode LDA. At the same time, we measure or infer the magnitude of the instantaneous velocity vector. This quantity is used to convert the time record to a spatial record consisting of elements of the instantaneous streakline through the MP as explained below. Since a spectrum must be computed as an average over many records (for example, by block averaging), we can view the measurement as illustrated in Fig. 18.2. This figure shows the spatial structures in a jet flow as recorded by an instantaneous Schlieren image. Block averaging would consist in an average of spectra from a number of consecutive time records corresponding to different streaklines passing through the MP.

Fig. 18.2

Schlieren image of the spatial structure of a free jet in air. Yellow: streaklines passing through the measurement point. From M. van Dyke: An album of fluid motion

The problem with a temporal record of the velocity measured as the fluid streams past the MP is that it does not accurately describe the spatial structure of the turbulence that is of interest to turbulence theory and to turbulence modeling. The small spatial structures are convected past the MP by the instantaneous large velocity. This convection of the small scales causes small spatial structures to appear at different temporal frequencies in the power spectrum depending on the instantaneous convection velocity. In other words, strong aliasing may distort the computed spectrum.

In the past, multiple attempts were made to pass from the temporal (or frequency) spectrum to a spatial (or wavenumber) spectrum, the simplest and best known one based on the so-called Taylor’s hypothesis (TH) [1] according to which the small scales are carried past the probe with the local mean velocity assumed to be so great that the small scales do not have time to change while being swept past the MP. This is also called the “frozen turbulence model”. However, this method works only for relatively low intensity turbulence (turbulence intensity defined as the rms fluctuating velocity relative to the mean velocity). In cases of high intensity turbulence, as, e.g., at the outer limits of the free turbulent jet or in atmospheric boundary layers, the TH is useless and even misleading as the instantaneous velocity fluctuates greatly. However, TH may work satisfactorily in some atmospheric flow measurements where the time/length scales of interest are sufficiently small [7].

TH has been examined in numerous studies, experimentally such as [8] and [9] and in a free round jet in [5, 10, 11]. Other studies assume theoretical models for the large-scale velocity structure such as [12]. Other sophisticated methods for transforming the time spectrum to a spatial one were presented in [2, 3, 13], but all relying on some assumption regarding the convection velocity.

Our method now means that we transform the time signal of, e.g., velocity component u _i recorded at the MP with a sampling interval Δ t and measured together with the magnitude of the three-dimensional velocity u =  | u | to a function of the length of the streakline element Δ s = uΔt of a fluid particle passing through the MP. In other words, we change the measurement from one using temporal sampling intervals to one using spatial sampling intervals, but still recorded at the MP. The method is illustrated in Fig. 18.3.

Fig. 18.3

The instantaneous streakline element through the MV

Δ s = uΔt is the (scalar) streakline element in the direction of the instantaneous velocity u. Δ s _i  = u _i Δ t is the projection of this element onto the i-direction. Thus, Δ s requires an independent measurement of the velocity magnitude u, whereas Δ s _i can be computed based on the measured velocity component u _i .

A similar approach was attempted by Pinton and Labbé [14] in a von Karman swirling flow in which there exists no steady well-defined mean velocity for applying TH. A “local” TH was therefore implemented, reminiscent of the streakline approach presented herein. However, the lengths of the spatial/temporal elements in this transformation were chosen to be of the order of the revolution time of the disc, or correspondingly the integral timescale of the flow. This is quite contrary to the approach in the current work, where we apply the instantaneous velocity magnitude and where we desire a high enough sample rate to resolve the smallest scales of interest.

2 Spatial Sampling

The method we present here requires in addition to the measurement of the desired velocity component u _i (t), the measurement of the magnitude of the velocity vector, u(t) =  | u(t) | at the MP. In the case of a HWA, this could be realized by a spherical probe mounted next to the HWA. In case of the burst-mode LDA, the particle transit time or the so-called residence time, Δ t _r , which is now measured in most commercial instruments, can be used to infer the velocity magnitude. u(t) can of course also be derived from a three-component LDA measurement. The residence time is used for “residence time weighting” (RTW) of time records thus compensating for the velocity–sample rate correlation or velocity bias and allowing the computation of bias-free time averages [15, 16]. However, the residence time in our temporal-to-spatial scheme is now also used for transforming the temporal spectrum to a spatial one.

2.1 Example: Regular Sampling (HWA)

Assume the measurement of a temporal record with the constant sampling interval Δ t, with velocity component samples u _i, n (t) and velocity magnitude u _n (t), as illustrated in Fig. 18.4 (this would apply to a hot-wire anemometer, HWA, able to measure both a velocity component and the velocity magnitude):

Fig. 18.4

Sampling the velocity time signal with constant temporal sample rate. The black curve displays the velocity magnitude while the red curve displays the measured single velocity component

We choose to use in the following the so-called direct method or “periodogram” to compute the spectrum, S _i (f), of the velocity component u _i , but the results apply to other computational methods as well:

$$\displaystyle{ S_{i}(f) = \frac{1} {T}\hat{u}_{i}(f)\hat{u}_{i}(f)^{{\ast}} }$$

(18.1)

where T is the length of the time record and $\hat{u}_{i}(f)$ is the Fourier transform of u _i (t). In case of the HWA the samples are equidistant, so we can use the fast Fourier transform (FFT):

$$\displaystyle{ \hat{u}_{i}(f) = \frac{T} {N}\sum _{n=1}^{N}e^{-i2\pi ft_{n} }u_{i,n},\quad \quad \mathrm{where}\quad \quad t_{n} =\sum _{ n'=1}^{n}\varDelta t = n\varDelta t. }$$

(18.2)

Knowing u _n , we can compute the corresponding length of the instantaneous streakline element through the MP in the time Δ t: Δ s _n  = u _n Δ t. We can then construct a spatial sampling record, illustrated in Fig. 18.5, from the spatial sampling intervals, Δ s _n :

$$\displaystyle{ s_{n} =\sum _{ n'=1}^{n}\varDelta s_{ n'} =\sum _{ n'=1}^{n}u_{ n'}\varDelta t }$$

(18.3)

Using this record, we can compute the spatial energy spectrum (we shall apply the term energy spectrum for the spectrum in the spatial domain) for the i’th velocity component, S _i (k), by the direct method:

$$\displaystyle{ S_{i}(k) = \frac{1} {L}\hat{u}_{i}(k)\hat{u}_{i}(k)^{{\ast}}, }$$

(18.4)

where L is the length of the spatial record and $\hat{u}_{i}(k)$ is the Fourier transform of u _i (s). Note that this is a one-dimensional spectrum as a function of the wave number k, which is a scalar.

Fig. 18.5

Spatial sampling when sampled with constant temporal sample rate. The black curve displays the velocity magnitude while the red curve displays the measured single velocity component

Since the samples are no longer equidistant, we cannot use the FFT, but must use the discrete Fourier transform (DFT):

$$\displaystyle{ \hat{u}_{i}(k) = \frac{N} {L} \sum _{n=1}^{N}e^{-i2\pi ks_{n} }u_{i,n}\quad \quad \mathrm{where}\quad \quad s_{n} =\sum _{ n'=1}^{n}u_{ n'}\varDelta t. }$$

(18.5)

The small-scale spatial structures are now displayed correctly and the spatial spectrum S _i (k) is now computed without the scrambling caused by the convection velocity.

2.2 The Method Applied to LDA

One might be a bit skeptical when it comes to applying the method to the burst-mode LDA. The LDA presents some unique signal processing challenges. Firstly, the LDA data arrive randomly with a sampling rate that contains a sampling rate—velocity correlation. Secondly, the random sampling results in random temporal sampling intervals corresponding to the interval between arrival times for the particles passing the laser beam.

Furthermore, the average data rate is often relatively low compared to the highest frequencies in the velocity signal. We then cannot convert a small constant temporal sampling interval, Δ t, to a variable spatial one, Δ s, as we did in the HWA case. The only information we have about the temporal sampling interval is the time between samples, and if this time is of the order of the temporal scales of the velocity, we cannot accurately compute the path length along a streamline simply by using the measured velocity magnitude and the time between samples. On the other hand, the measured residence time allows us to estimate the magnitude of the velocity vector in addition to the measured velocity component that we get from the Doppler modulation.

The temporal power spectrum in the case of the burst-mode LDA is again given by (using the direct method):

$$\displaystyle{ S_{i}(f) = \frac{1} {T}\hat{u}_{i}(f)\hat{u}_{i}(f)^{{\ast}}, }$$

(18.6)

where T is the length of the time record and $\hat{u}_{i}(f)$ is the Fourier transform of u _i (t).

The samples are not equidistant, so we must use the DFT (using residence time weighted formulas):

$$\displaystyle{ \hat{u}_{i}(f) = \frac{N} {T} \sum _{n=1}^{N}e^{-i2\pi ft_{a,n} }\varDelta t_{r,n}u_{i,n}/\sum _{n=1}^{N}\varDelta t_{ r,n} }$$

(18.7)

where Δ t _r, n is the measured residence time and t _a, n is the measured arrival time.

We now use the known diameter of the measurement volume, d _MV , and the measured residence time to get a measure of the velocity magnitude: u _n  = d _MV ∕Δ t _r, n . We can compute the corresponding length of the instantaneous streakline passing through MP in the time Δ t _n  = t _a, n − t _a, n−1: Δ s _n  = u _n Δ t _n . We then construct a spatial sampling record from the temporal random sampling intervals, Δ t _n :

$$\displaystyle{ s_{n} =\sum _{ n'=1}^{n}\varDelta s_{ n'} =\sum _{ n'=1}^{n}u_{ n'}\varDelta t_{n'} =\sum _{ n'=1}^{n}\left (d_{ MV }/\varDelta t_{r,n'}\right )\left (t_{a,n'} - t_{a,n'-1}\right ). }$$

(18.8)

Using this record, we can compute the spatial energy spectrum for the i’th velocity component, S _i (k), by the direct method:

$$\displaystyle{ S_{i}(k) = \frac{1} {L}\hat{u}_{i}(k)\hat{u}_{i}(k)^{{\ast}}, }$$

(18.9)

where L is the length of the spatial record and $\hat{u}_{i}(k)$ is the Fourier transform of u _i (s). Note that this is again a one-dimensional spectrum as a function of the wave number k, which is a scalar.

Since the samples are not equidistant, we cannot use the FFT, but must use the DFT:

$$\displaystyle{ \hat{u}_{i}(k) = \frac{N} {L} \sum _{n=1}^{N}e^{-i2\pi ks_{n} }\varDelta t_{r,n}u_{i,n}/\sum _{n=1}^{N}\varDelta t_{ r,n} }$$

(18.10)

where L is the record length. The small-scale spatial structures are now again displayed correctly and the spatial spectrum S _i (k) is now computed without distortion due to the convection velocity.

3 Tests with a Computer Generated Large Eddy

To illustrate the method, we generate a spatial structure consisting of a large vortex or eddy and superimpose on this convection velocity a small, harmonically oscillating spatial structure that is carried along with the velocity of the large eddy. The center of the large eddy travels with a constant velocity, u _c  = 2 m s⁻¹, along the flow direction. The geometry is shown in Fig. 18.6.

Fig. 18.6

The figure shows the large eddy with velocity u _l (red curve) plus the small harmonic oscillator u _s . The small harmonic oscillation is described by a spatial velocity function: u _s (s) = u _s, 0cos(2π s∕λ), where λ is the spatial period along s of the small structure

3.1 HWA Signal

We compute the power spectrum of a 1 s long time record u _i (t), modulated with a spatial period of 0. 005 m corresponding to a spatial frequency of 200 m⁻¹ and sampled with a sample rate of 1000 sps. This spectrum is shown in Fig. 18.7a. The original spatial modulation of 200 m⁻¹ appears scrambled in the temporal spectrum due to the changing convection velocity. Assuming measurement of the velocity magnitude at the MP and using the conversion described above, we obtain a spatial spectrum S _i (k) as shown in Fig. 18.7b. The original spatial harmonic of 200 m⁻¹ has been restored.

Fig. 18.7

Spectra of the large eddy and the small spatial modulation measured with HWA. PST: Temporal spectrum. The spatial 200 m⁻¹ modulation appears as aliased frequencies in the temporal spectrum. PSR: Using the conversion method the spatial spectrum is restored. The slight broadening of the 200 m⁻¹ peak is due to the low frequency content of the large eddy

3.2 LDA Signal

We assume a uniform particle distribution in space. The LDA probe will see a random temporal sample rate correlated with the velocity magnitude. However, the overall result of the temporal-to-spatial conversion is similar to the HWA result, see Fig. 18.8.

Fig. 18.8

Spectra of the large eddy and the small spatial modulation measured with LDA. PST: Residence time weighted temporal power spectrum. The spatial 200 m⁻¹ modulation is aliased in the temporal spectrum. PSR: Using the conversion method the spatial energy spectrum is restored. The slight broadening of the 200 m⁻¹ peak is due to the low frequency content of the large eddy

4 Tests with Computer Generated LDA Data with a Von Karman Spectrum

The computer generated data has been described in [17]. Briefly, the random spatial samples were grabbed from a high rate primary velocity record with a Von Karman power spectrum by a Poisson process. Since we are sampling the velocity in space and assuming the particles to be uniformly distributed in space, the Poisson process is not modulated by velocity, and there is no velocity–sample rate bias. The Von Karman spectrum was chosen to model as closely as possible the measured jet spectrum described below:

$$\displaystyle{ S_{vK}(f) = \frac{1} {62.5} \cdot \frac{1} {\left [1 + (f/45)^{2}\right ]^{5/6}} \cdot \exp \left [-(f/2500)^{4/3}\right ]. }$$

(18.11)

The integral time scale of this process can be found from l = S(0)∕2var(u).

We now assume that the Von Karman velocity fluctuations are convected through the LDA MV by 3-D Gaussian large-scale eddies. The turbulence intensity for the complete fluctuating convection velocity signal was 54 %. The resulting randomly sampled time record results in an aliased time spectrum (dark blue) as shown in Fig. 18.9 LHS.

Fig. 18.9

Spectra of the Von Karman turbulence convected past the LDA MV by a large low frequency Gaussian fluctuation. Left: blue: The temporal spectrum. Right: yellow: The original Von Karman model spectrum. Red: The spatial spectrum restored. Blue: The spatial spectrum restored by the conventional Taylor’s hypothesis

The mean velocity was 1 m s⁻¹ and the record length was 1 s, which allows us to use the same scale on the abscissa for both temporal and spatial spectra. The figure shows that the method restores the Von Karman spectrum even in the presence of large-scale low frequency 3-D fluctuations. It also shows that the use of Taylor’s hypothesis in this case with a 50 % turbulence intensity does not adequately convert the time record to a spatial one.

5 Measurements

We now try our method on LDA measurements in a free, round, turbulent jet in air. The measured velocity is the streamwise component. This experiment is described in detail in [16]. The jet data for these measurements were: Jet exit diameter: 10 mm, measurement location: 30 diameters downstream, 0, 13, 26, 39, 52 mm off axis, center mean velocity: 6. 2 m s⁻¹, center line turbulence intensity: 7. 5 %, Re = 20, 000, integral time scale 0. 0048 s. The active measurement volume was 75 × 700 μm and the average data rate was approx. 6400 s⁻¹. The scales are estimated as: Kolmogorov scale = 53 μm, Taylor scale = 2. 2 mm at the center line. 4. 000. 000 data points were used.

Figure 18.10 shows the measured temporal spectra. The power spectra are shifted towards higher frequencies as the mean velocity increases. The total power also increases as the local mean velocity increases, but these spectra are normalized to the same values at low frequency in order to be able to compare the shape of the spectra. The spectra show high similarity for the high frequency fluctuations in agreement with the qualitative impression from the Schlieren photograph in Fig. 18.1. The dip at high frequency is due to dead time [18, 19], which has not been removed by deconvolution, as is otherwise possible, see [17].

Fig. 18.10

Power spectra of jet measurements at different off-axis positions. From heavy blue to light blue: off-axis position 0, 13, 26, 39, 52 mm

After application of our conversion method, we find the spatial spectra shown in the left graph of Fig. 18.11. These should be compared to the true spatial (PIV) spectra in Fig. 18.11 right. The red color shows power spectra based on the spatial record obtained by our method. The blue curve shows power spectra based on Taylor’s method. The spectra based on the method using the measured magnitude of the velocity collapse to nearly overlapping curves whereas Taylor’s method fails for the 26, 39, and 52 mm off-axis positions. The turbulence intensity at these positions are 23 %, 65 %, and 420 %, respectively.

Fig. 18.11

Left: Restored spatial spectra. Red: Our method. Off-axis position heavy red to light red: 0, 13, 26, 39, 52 mm. Blue: Taylor’s hypothesis. Off-axis position heavy blue to light blue: 26, 39, 52 mm. Right: Spatial spectra from homogeneous directions in the jet using PIV [20]. From light blue to heavy blue: 0, 13, 26, 39, 52 mm. When normalizing the spectra by their respective energy, they collapse onto the same curve

6 Discussion

Apart from the practical questions related to this method (the possibility for measuring the magnitude of the 3-D velocity vector with a HWA and with the LDA), a number of more principal questions deserve a further investigation:

• How do we interpret the first-order statistics (e.g., mean values) computed from the data with and without residence time correction?

• There are a number of different ways of measuring and calculating the turbulent power spectrum. How do they relate to the classical definitions of turbulent power spectra and to the classical kinetic energy spectrum?

6.1 Practical Considerations

It is well known that the HWA has a limited acceptance angle for the flow incident on the wire. It may well be that in practice the allowed turbulence intensity for the HWA is so low that the ordinary Taylor method is adequate and that a measurement of the velocity magnitude is unnecessary. However, the method applies to other measurements as well, e.g., fluctuating temperature or concentration. It might be feasible to construct a probe with a small omnidirectional (spherical) sensor and another sensor, e.g., a temperature sensor, located close by. It would then be possible to convert a temporal temperature or concentration measurement to the spatial domain and remove the aliasing effect connected with a power spectrum in the time domain.

As we have demonstrated above, it is feasible for an LDA to measure both a velocity component (by the ordinary Doppler effect) and the velocity magnitude (by means of the measured residence time). We can then convert a time record of a flow with unlimited turbulence intensity (even a flow with no mean convection velocity) to a spatial record and compute a spatial energy spectrum. The practical concerns related to this method arise in connection with especially two effects: Firstly, the random sampling causes the usual additional noise known from randomly sampled records. However, an additional concern is related to the measured random time between samples, which must be used to estimate the spatial sampling intervals from the temporal intervals. It is obvious that if the time between samples is much longer than the typical time scales of the velocity, information about the spatial record between measurements is lost as only a zeroth order interpolation replaces the true streakline element. The method thus requires a relatively high average sample rate. This, however, is already a criterion for obtaining high quality power spectra c.f. [15, 16]. Secondly, the measured residence time fluctuates greatly due to the different particle trajectories through the MV. However, the same concern applies to other LDA measurements that use the residence time, and experience shows that this noise does not add appreciably to the noise already present from the random sampling. Both noise effects are reduced by the block averaging that is necessary anyway to obtain low noise spectra.

6.2 Interpretation of First-Order Statistics

We consider here the mean values computed from the data as other statistical quantities must be interpreted the same way. The straightforward way of measuring the mean velocity is to simply use an arithmetic mean:

$$\displaystyle{ \mathrm{Temporal\,\,\,mean\,\,\,value,\,\,\,HWA:}\quad \quad \overline{u}_{i} = \frac{\sum _{n=1}^{N}u_{i,n}} {\sum _{n=1}^{N}1} = \frac{1} {N}\sum _{n=1}^{N}u_{ i,n}. }$$

(18.12)

This gives the correct temporal mean value for a regularly sampled HWA signal (however, the spatial structures are distorted due to the convection effect). When we apply our conversion method to the regularly sampled time signal, the resulting spatial record shows what we may call an “inverted velocity bias.” This is because the measured velocities will now be close together on the spatial record where the velocity magnitude is low and further apart where the velocity magnitude is high. However, even if the velocities in this spatial record are unevenly spaced, they are the same values as in the time record, and the velocity mean value is still the arithmetic mean:

$$\displaystyle{ \mathrm{Spatial\,\,\,velocity\,\,\,mean:}\quad \quad \langle u_{i}\rangle = \frac{\sum _{n=1}^{N}u_{i,n}} {\sum _{n=1}^{N}1} = \frac{1} {N}\sum _{n=1}^{N}u_{ i,n}. }$$

(18.13)

The small spatial structures, however, are now spaced correctly on the spatial record.

The mean value computed from a randomly sampled LDA signal leads to an overweight of high velocity samples (“velocity bias”). As shown theoretically in [16] and by computer generated data in [17], weighting by the measured residence time gives correct time mean values for any sample rate:

$$\displaystyle{ \mathrm{Temporal\,\,\,velocity\,\,\,mean\,\,\,value:}\quad \quad \overline{u}_{i} = \frac{\sum _{n=1}^{N}\varDelta t_{r,n}u_{i,n}} {\sum _{n=1}^{N}\varDelta t_{r,n}}. }$$

(18.14)

The question is now: When this temporal record is converted to a spatial record consisting of a sum of spatial intervals of the instantaneous streakline through the MP, but the measured velocities remain the same, how do we then compute the velocity mean value? Since the measured velocities are now spaced by the original spatial distribution, which we assume uniform in space, it is tempting to consider the simple arithmetic average as a measure of the velocity mean value. However, the measurement is still based on the biased sampling in time, and therefore the mean value must still be corrected with the measured residence times. The arithmetic mean of the measured velocities must instead be interpreted as a measure of the volume flow through the MV. This is easily seen by reference to Fig. 18.3 above. The streakline element Δ s multiplied by the MV cross section is the volume of fluid carried through the MV in the time element Δ t. Thus the sum

$$\displaystyle{ \overline{\rho } = \frac{\sum _{n=1}^{N}u_{i,n}\varDelta t} {\sum _{n=1}^{N}\varDelta t} = \frac{\sum _{n=1}^{N}\varDelta s_{i,n}} {T}. }$$

(18.15)

is a measure of the average volume flow density through the MV. To get the correct velocity mean we must again weigh the data in the spatial record by the residence time, Δ t _r :

$$\displaystyle{ \mathrm{Spatial\,velocity\,mean\,value:}\quad \quad \langle u_{i}\rangle = \frac{\sum _{n=1}^{N}\varDelta t_{r,n}u_{i,n}} {\sum _{n=1}^{N}\varDelta t_{r,n}}. }$$

(18.16)

We thus see a principal difference between the regularly sampled time signal from, e.g., a HWA and the random data of a burst-mode LDA. We can say that the HWA samples uniformly in time whereas the LDA samples uniformly in space. LDA data measured in the time domain must be corrected with residence time, also when the record is converted back to a spatial record.

6.3 Interpretation of the Measured Spectra

Again, we must distinguish between the regularly sampled time signal from a HWA and the spatially sampled LDA. Taking first the HWA, it is generally accepted that the small spatial velocity structures are swept past the stationary probe by the large convective velocity components. The distinction between small-scale structures and large-scale convective structures is somewhat artificial, however. In reality, the “sweeping” is done by the total instantaneous velocity vector. In any case, the result is that the temporal power spectrum is aliased; the small structures are smeared out over a large temporal frequency domain, especially when the turbulence intensity is greater than about 40 % as we have seen in the examples above. To analyze the computed spectra further, we must distinguish between the various velocities and restored spatial records. The desired spectrum is for the measured velocity component u _i (or possibly the temperature fluctuations or concentration fluctuations). As we have seen above, we can construct two spatial records: One based on the measurement of the magnitude of the three-dimensional velocity vector, u =  | u | , and one based on the measured velocity component, u _i . The energy spectra computed for the velocity component, u _i , based on these two spatial records, s and s _i are both one-dimensional. The first is given by

$$\displaystyle{ \begin{array}{ll} \mathrm{1a:}&\quad \quad S_{i}(k_{i}) = \frac{1} {L}\hat{u}_{i}(k_{i})\hat{u}_{i}(k_{i})^{{\ast}}\quad \mathrm{with}\quad \hat{u}_{ i}(k_{i}) = \frac{N} {L} \sum _{n=1}^{N}e^{-i2\pi k_{i}s_{i,n}}u_{ i,n} \\ &\quad \quad \mathrm{and}\quad s_{i,n} =\sum _{ n'=1}^{n}u_{i,n'}\varDelta t.\end{array} }$$

(18.17)

where k _i is a scalar wave number in the direction of the measured velocity component u _i . The other is expressed by

$$\displaystyle{ \begin{array}{ll} \mathrm{2a:}&\quad \quad S_{i}(k) = \frac{1} {L}\hat{u}_{i}(k)\hat{u}_{i}(k)^{{\ast}}\quad \mathrm{with}\quad \hat{u}_{ i}(k) = \frac{N} {L} \sum _{n=1}^{N}e^{-i2\pi ks_{n}}u_{ i,n} \\ &\quad \quad \mathrm{and}\quad s_{n} =\sum _{ n'=1}^{n}u_{n'}\varDelta t. \end{array} }$$

(18.18)

where k is a scalar wavenumber in the direction of the instantaneous streakline. Before we discuss these spectra any further, we shall consider the LDA case. The corresponding LDA energy spectra are given by (residence weighted formulas, see [15]):

$$\displaystyle{ \begin{array}{ll} \mathrm{1b:}&\quad \quad S_{i}(k_{i})= \frac{1} {L}\hat{u}_{i}(k_{i})\hat{u}_{i}(k_{i})^{{\ast}}\quad \mathrm{with}\quad \hat{u}_{ i}(k_{i})=\frac{N} {L} \sum _{n=1}^{N}e^{-i2\pi k_{i}s_{i,n}}\varDelta t_{ r,n}u_{i,n}/\sum _{n=1}^{N}\varDelta t_{ r,n} \\ &\quad \quad \mathrm{and}\quad s_{i,n}=\sum _{n'=1}^{n}u_{i,n'}\varDelta t_{n'}.\end{array} }$$

(18.19)

and

$$\displaystyle{ \begin{array}{ll} \mathrm{2b:}&\quad \quad S_{i}(k) = \frac{1} {L}\hat{u}_{i}(k)\hat{u}_{i}(k)^{{\ast}}\quad \mathrm{with}\quad \hat{u}_{ i}(k) = \frac{N} {L} \sum _{n=1}^{N}e^{-i2\pi ks_{n}}\varDelta t_{ r,n}u_{i,n}/\sum _{n=1}^{N}\varDelta t_{ r,n} \\ &\quad \quad \mathrm{and}\quad s_{n} =\sum _{ n'=1}^{n}u_{n'}\varDelta t_{n'}. \end{array} }$$

(18.20)

There is a possibility for one more energy spectrum based on the velocity magnitude or rather on the 3-D velocity vector in the direction of the instantaneous streakline through the MP:

$$\displaystyle{ \begin{array}{ll} \mathrm{3:}&\quad \quad S(k) = \frac{1} {L}\hat{u}(k)\hat{u}(k)^{{\ast}}\quad \mathrm{with}\quad \hat{u}(k) = \frac{N} {L} \sum _{n=1}^{N}e^{-i2\pi ks_{n}}\varDelta t_{ r,n}u_{n}/\sum _{n=1}^{N}\varDelta t_{ r,n} \\ &\quad \quad \mathrm{and}\quad s_{n} =\sum _{ n'=1}^{n}u_{n'}\varDelta t_{n'}. \end{array} }$$

(18.21)

Note: This is again a one-dimensional spectrum; the measured 3-D velocity vector is always in the direction of the streakline.

We now suggest the following interpretation of the high wavenumber part of these spectra, assuming that the small spatial structures are isotropic. We can think of the small scales as an isotropic superposition of plane waves. Expressions 1 and 2 above are spatial energy spectra of the measured velocity component, u _i . The “a” and “b” indices refer to the HWA and LDA cases, respectively. Expressions 1a and 1b are converted by the use of the magnitude of the measured velocity component, whereas 2a and 2b use the measured magnitude of the instantaneous 3-D velocity vector. Expression 3 is the energy spectrum of the measured 3-D velocity vector magnitude using this quantity also for the conversion process.

Spectrum 1 is then the energy spectrum of the measured velocity component in the direction of the i-axis on a spatial record, which is composed of the projection of the streakline element onto the i-direction. We consider this equivalent to the usual one-dimensional energy spectrum of the i-component of the velocity along a line through the MP in the i-direction, assuming local homogeneity for the flow along the measurement line. As the plane wave decomposition of the small-scale structure is projected onto the i-direction, the projection of the plane waves onto the i-axis will appear at an increased wavelength, and the one-dimensional spectrum will be direction-aliased. Spectrum 2 is again the energy spectrum of the measured i-component of the velocity, but in this case referred to the spatial record composed of steak line elements in the direction of the 3-D flow velocity. Here we measure the transport along the streakline of the i-component of the small-scale velocity structures through the MP. This one-dimensional spectrum is also aliased. Finally, we associate the third spectrum with the energy spectrum of the velocity magnitude along a spatial record, which is always in the same direction as the velocity vector. This must mean that the final wave front of the spatial structures is always normal to the direction of the streakline element, and that the streakline element and the velocity vector are always co-parallel. There can be no velocity component normal to the streakline. Spectrum 3 must therefore express the total kinetic energy. This spectrum is also one-dimensional and will therefore again be aliased. Note, however, that we obtain the total kinetic energy by a single measurement; not as in the classical case as a sum of energy spectra along three orthogonal coordinate axes.

We have computed two spatial spectra 2a and 2b from the measured jet LDA data. Figure 18.12 shows the result for an off-axis position of 39 mm, indicating that the spectrum based on a projected k-vector component is shifted towards higher frequencies as compared to the one based on the 3-D streakline.

Fig. 18.12

LDA measurements of power spectrum, direct method, uniform random sampling in space, 100 record block average. Red: Reconstructed spatial record s. Light red: Reconstructed spatial record s _x

Figures 18.13 and 18.14 display a comparison between the 2b and 3 spectra, the difference being that the spectra are based on a single velocity component and the full velocity vector, respectively. The left column shows the temporal spectra and the right column the spatial ones. At the jet centerline, these two spectra are expected to display the same behavior due to the (relatively) low turbulence intensities, which is seen to be true for both the temporal and spatial spectra. The dip seen in the spectra arise due to dead time effects. Interestingly, this concurrence still persists as far out as 39 mm (1. 5 jet half-widths) from the jet centerline.

Fig. 18.13

Comparison of spectra 2b and 3. Jet measurements, y = 0 mm. LHS: Temporal spectra. RHS: Spatial spectra. Light color: Case 2b, Dark color: Case 3

Fig. 18.14

Comparison of spectra 2b and 3. Jet measurements, y = 39 mm. LHS: Temporal spectra. RHS: Spatial spectra. Light color: Case 2b. Dark color: Case 3

7 Linking to Big Data

We propose the following interpretations of the computed spectra in Sect. 18.6.3: The spatial isotropic small-scale spectra obtained by the use of the measured instantaneous velocity magnitude are interpreted as the classical one-dimensional spectrum of the measured velocity component (spectrum 2b) whereas we interpret spectrum 3 as the one-dimensional total kinetic energy spectrum. Interestingly, this last spectrum is obtained by just a single measurement and not as the sum of three measurements along orthogonal axes.

The results of our investigations are relevant to the “Big Data” theme of this conference by helping to improve the accuracy of experimental databases, in particular concerning the important question as to how energy is cascaded from the energy creating eddies to the high frequency dissipative scales. Direct solution of the fundamental equations has proven infertile up to now, and the future response to fluid mechanics problems may be a combination of physical guide lines and a huge database of experiments, all under the auspices of a “Google-like” artificial intelligence. Obviously, such solutions require large databases of accurate experimental results.

8 Conclusion

A novel method for converting temporal turbulent flow measurement records into spatial ones, bypassing the adverse fluctuating convection velocity effect, is proposed. The method is based on the principle of converting time records into spatial piecewise streakline records using the instantaneous measured velocity magnitude. By measuring along a streakline in time from a fixed point in space, the seldom fulfilled requirement of statistical homogeneity can be replaced by the considerably more relaxed requirement of statistical stationarity. The method requires that the flow scales of interest are well resolved in space/time, in particular the smallest scales that are expected to deviate most from the true spatial representation. Unless this temporal-spatial conversion is performed correctly, the structures of the original signal are scrambled in its transformed counterpart. This was exemplified using computer simulations of a large eddy with a superimposed high spatial frequency modulation, where the original signal could be restored using the proposed method. Computer generated data from a von Karman spectrum model of turbulence were used to demonstrate the kind of adverse energy scrambling effects that are to be expected in a real turbulent flow measurement and how well the proposed method is able to alleviate them. The method was finally demonstrated on measured data from an axisymmetric turbulent jet measured at different distances from the centerline. This is a direct test of the method since the turbulence intensity increases dramatically with radial distance from the jet center, a requirement well known to be challenging for the classical Taylor’s hypothesis not least due to the fluctuating convection velocity effect. This is also evident in the present comparison between this and our proposed method, where the latter displays exactly the same behavior as the true spatial spectra measured with PIV along homogeneous flow directions.

We propose interpretations of the computed spectra. The spatial isotropic small-scale spectra obtained by the use of the measured instantaneous velocity magnitude are interpreted as the classical one-dimensional spectrum of the measured velocity component (Spectrum case 2b) and whereas we interpret Spectrum case 3 as the one-dimensional total kinetic energy spectrum. Interestingly, this last spectrum is obtained by just a single measurement and not as the sum of three measurements along orthogonal axes.