Measurement of the fluctuating temperature field in a heated swirling jet with BOS tomography

Abstract

This work investigates the potential of background-oriented schlieren tomography (3D-BOS) for the temperature field reconstruction in a non-isothermal swirling jet undergoing vortex breakdown. The evaluation includes a quantitative comparison of the mean and phase-averaged temperature field with thermocouple and fast-response resistance thermometer as well as a qualitative comparison between the temperature field and the flow field obtained from particle image velocimetry (PIV). Compared to other temperature-measuring techniques, 3D-BOS enables non-invasive capturing of the entire three-dimensional temperature field. In contrast to previous 3D-BOS applications, the present investigation makes use of the special character of the flow, which provides a global instability that leads to a rotational symmetry of the jet. Additionally, the rotational motion of the jet is used to obtain a tomographic reconstruction from a single camera. The quality of 3D-BOS results with respect to the physical setup as well as the numerical procedure is analyzed and discussed. Furthermore, a new approach for the treatment of thin occluding objects in the field of view is presented.

Appendix: Time series statistics

The investigated flow exhibits two distinct timescales that influence the convergence of the measured mean values and the related confidence of the measured temperatures. These are the high-frequency PVC oscillations and slow changes of the velocity field that alter the position of the vortex breakdown bubble (Rukes et al. 2015). As stated in the main text, the PVC oscillations are not present over the entire time of the measurement, but they rather appear and disappear intermittently. This goes in hand with movements of the breakdown bubble. To provide reliable mean values, the timescales of both motions have to be be captured sufficiently.

The different timescales of the flow are discussed by means of pressure measurements at the nozzle. A continuous measurement over 1 h is used to investigate the spectral characteristics of the flow. In Fig. 21, the averaged power spectral density is displayed using Welch’s method, where the measurement is split into 20 segments of 360 s with 50% overlap and the squared magnitude of the Fourier transform of the segments is averaged. The spectral density from the average of all eight sensors is shown, which is assumed to represent the slow drift of the velocity field. Moreover, the first azimuthal Fourier mode that represents the PVC dynamic is shown [see Eq. (2)] as well as its magnitude, which characterize the amplitude modulation of the PVC. The graph shows that the flow exhibits several characteristic timescales, which can be separated into two phenomena. The PVC oscillation gives a clear peak at 12Hz in the $p_{\mathrm {PVC}}$ spectrum, whereas the $p_0$ and magnitude of $|p_{\mathrm {PVC}}|$ has similar trends with characteristic humps at $1\mathrm {~Hz}$ and $0.01 \mathrm {~Hz}$. The similarity between the two curves confirms the assumption that the low-frequency changes of the mean flow couple with the amplitude modulation (intermittency) of the PVC. Higher-frequency peaks of the $p_0$ spectrum are related to acoustic resonances of the flow duct upstream of the nozzle.

Fig. 21

Power spectral density from pressure measurements an the nozzle. The average pressure of all eight sensors $p_0$ as well as first azimuthal harmonic $p_{\mathrm {PVC}}$ and its magnitude $|p_{\mathrm {PVC}}|$ are shown

An inspection of the histogram (not shown) of $p_0$ low-pass filtered at $0.1 \mathrm {~Hz}$ reveals that the slow fluctuations exhibit a bimodal behavior, whereas the fluctuation at $1 \mathrm {~Hz}$ are rather normally distributed. The PVC frequency is again much larger and, therefore, plays only a minor role for the convergence of time averaged temperatures. To give some more detail, the time series of a RTD measurement at $x/D=0.33$ on the centerline ($y=z=0$) is investigated. The series is just one point of the data that is already investigated above, which were acquired for 60 s. The averaging time with respect to the largest timescales already suggest some deficits that will be detailed now. In Fig. 22, a part of the time series together with the spectral density and the probability density from the entire series is given. In this case, the spectral density is averaged for overlapping segments of 10 s. The probability density is given as a normalized histogram with bin sizes of $2.8 \mathrm {~K}$.

Fig. 22

Different representations of the RTD measurement time series at $x/D=0.33$ and $y/D=0$: part of the raw time series (a), averaged power spectral density (b) and probability density of the temperature (c)

The time series (Fig. 22a) shows some states where the sensor measures the constant jet temperature ($20 \mathrm {~^{o}C}$) and states where a heated fluid passes by the sensor. The switching time between both states seems random; however, at the beginning of the series, the ’cold’ states appear more often than the ’hot’ ones. The power spectral density (Fig. 22b) shows a continuous decay from 1 to $100 \mathrm {~Hz}$. There is no peak at the PVC frequency because the helical mode does not cause any fluctuations on the centerline (compare Fig. 15). At frequencies below 0.2, there is a strong increase in amplitude that suits the $0.01 \mathrm {~Hz}$ dynamic found in the pressure spectrum (Fig. 21). Unfortunately, the shorter measurement duration does not allow the resolution of the entire hump. The probability density (Fig. 22c) clearly shows a bimodal shape that corresponds to the two states identified in the time series. It is also obvious that the distribution of temperatures is far from a normal distribution.

From the presented time series, we conclude that the flow randomly switches between two states, where the temperature at the investigated position is either ’cold’ or ’hot’. The average switching time between both states appears to be rather random but approximately in the range between 0.1 and 1 s. This corresponds to the higher frequency of $1 \mathrm {~Hz}$ found in the pressure spectrum. However, the properties of this switching process are again altered by another bi-stability with time scales in the order of 10 to 100 s. This second process alters the probability of the flow to be either in the ’cold’ or ’hot’ state. With respect to the bimodal probability distribution (Fig. 22c) this corresponds to a change of the relative height of the two peaks. However, this has large impact on the average temperature measured at this point. In consequence, the averaging times for the RTD measurement have been too short. Nevertheless, it was not possible to extend the averaging time without extending the measurement time for the entire filed over several days. This again would include disturbances and changes of ambient parameters with daily timescales that we were trying to avoid. Another escape that we used for the processing of the phase-averaged data was the restriction of the evaluation to flow states where a strong PVC is present. Therefore, the low-frequency bi-stability is eliminated from the analysis by restricting to only one of the states that is present in the flow. An indication that the convergence is improved with this procedure is that the phase-averaged temperatures show similar coincidence as the averaged temperatures (compare Figs. 13 and 14) although only 10 to 20 % of the data is used for the phase-averaged evaluation.

At this point, a general advantage of the 3D-BOS method in comparison to the punctual consecutive measurement of the temperature field becomes obvious. No matter how long the measurement time of a 3D-BOS measurement is, the entire volume is always recorded at the same ensemble of flow states. If single points are measured one after another, every point has to be measured long enough to make sure that all flow states have appeared equally often. Therefore, 3D-BOS measurements will never show the erratic zigzag pattern that are commonly observed in profile measurements of turbulent flows (see Fig. 15). However, the smooth 3D-BOS profiles might also be elusive in terms of the actual convergence of the mean values as shown below. In the present investigation, the focus is not mainly the actual temperature but rather the relative distribution of heated fluid. In this case, the 3D-BOS is very valuable as it allows a tremendous reduction of measurement time.

To give an approximate confidence for the averaged temperatures, we further investigate the rate of convergence for the time series shown above. Consulting an time-series statistics textbook (Schlittgen and Streitberg 2001), we find that simple rules that apply for signals with Gaussian distribution and uniform spectral density do not apply. Therefore, convergence rate proportional to variance over number of samples is not expected. More generally, the variance of the mean temperature over N samples

$$\begin{aligned} \overline{T}_N = \frac{1}{N}\sum _{t=1}^N{T_t} \end{aligned}$$

(15)

can be described as

$$\begin{aligned} \mathrm {Var}(\overline{T}_N) = \frac{1}{N^2}\left[ N\gamma _0 + 2\sum _{\tau =1}^{N-1}{(N-\tau )\gamma _{\tau }}\right] , \end{aligned}$$

(16)

where $\gamma _{\tau }$ is the covariance

$$\begin{aligned} \gamma _{\tau } = \mathrm {Cov}[T_t,T_{t+\tau }] \approx \frac{1}{N}\sum _{t=1}^N{(T_t-\overline{T}_N)(T_{t+\tau }-\overline{T}_N)}. \end{aligned}$$

(17)

Here, the covariance can also only be estimated from the measured data what is especially problematic for large $\tau$ where the finite signal must be zero padded to evaluate the sum. The variance of the mean temperature $\overline{T}_N$ of the signal shown above (Fig. 22) for increasing averaging times is given in Fig 23. There, it is visible that the variance initially converges with a power law ($\propto 1/{\sqrt{N}}$) and later decays more strongly. The final decay is most probably caused by the drop of the estimated covariance for large N due to the zero padding. The increase of the spectral density at the lowest frequencies would actually cause a flatter decay. To give a reference, a flat spectrum (white noise) causes a decay with 1 / N, whereas a spectrum with density proportional to $1/f^2$ (red noise) causes zero decay. Therefore, the final decay is ignored and the value of the trend line at 60-s averaging time is taken as the estimate for the mean variance. This gives a variance of $\mathrm {Var}(\overline{T}_N) = \sigma ^2 = 19 \mathrm {~K^2}$.

Fig. 23

Convergence of the variance of $\overline{T}_N$ for increasing averaging times $t = N\Delta t$ of time-series from Fig. 22. The variance is calculated explicitly among the mean values from segments of the time series and according to Eq. (16). A power law trend line is fitted to the initial part of the curve

Finally, the confidence interval is usually given as the limits between which the mean value lies with 95% probability. Therefore, a certain shape of the probability distribution must be assumed. The ideal case would be an Gaussian distribution where the 95% probability limit corresponds to two times the standard deviation $2\sigma = 8.7 \mathrm {~K}$. The distribution in Fig. 22c) indicates a strong deviation from a normal distribution. To allow an estimate of confidence interval for a more general distribution Chebyshev’s inequality is used, which reads (Schlittgen and Streitberg 2001)

$$\begin{aligned} P[|X-\mu |>\sigma \cdot k] \le \frac{1}{k^2}, \end{aligned}$$

(18)

where P indicates the probability, $k>0$ and $\mu$ is the mean of a random distribution X with variance $\sigma ^2$. In order for the probability to be smaller than 5%, $k=\sqrt{1/0.05}\approx 4.5$. Hence, the confidence limit is $4.5\sigma = 19.5 \mathrm {~K}$. According to this rough estimate of the confidence interval, the mean temperature at $x/D=0.33$ is determined with an accuracy between $\pm 10$ and $\pm 20 \mathrm {~K}$, depending on the probability distribution. This is far beyond the inaccuracies due to sensor and digitizer calibration, which are in the range of $\pm 1 \mathrm {~K}$. However, the values are consistent with the intuitive estimation of the convergence according to the deviation of the measurements in Fig. 13. Since the confidence intervals can only be roughly estimated, we did not compute them for every position to add error bars to the graphs. This appendix should rather give an indication for the magnitude of the confidence interval.