PIV error sensitivity analysis for free turbulent flow characterization

Transcription

1 Lisbon, Portugal, July, 014 PIV error sensitivity analysis for free turbulent flow characterization José Nogueira *, Roberto Jiménez, Mathieu Legrand, Antonio Lecuona Department of Thermal and Fluid Engineering, Universidad Carlos III de Madrid, Leganes, Spain * correspondent author: goriba@ing.uc3m.es Abstract In the state-of-the-art PIV, image sensors are growing in size and PIV algorithms are increasing in spatial resolution capabilities. These technological advances allow for the simultaneous measurement of increasingly larger spatial scales ranges in a single PIV snapshot. To take fully advantage of the available spatial scales range, PIV setup constraints establish a coupling between the laser sheet thickness and the time between laser pulses. This condition did not apply with older technologies. The resulting value for the optimum laser sheet thickness is in a range where the superposed flow information in the direction of the imaging sensor introduces special characteristics errors at the small scale limit. This may compromise the measurement of flow quantities like dissipation and turbulent kinetic energy (tke), for which the contribution of the small and medium scales may be relevant. Within this scenario the assessment of the smallest scale resolved in the PIV measurement is necessary to specify the measurement error. This paper proposes the use of the classical second order structure functions of the flow and the variation of the laser sheet thickness to define error assessment protocols for these small scales. The results confirm that the laser sheet thickness spuriously increases the measured amplitude of velocity fluctuations at small scales. They also indicate that the magnitude of the error is not negligible. The use of the proposed tools seems promising, although further work is advisable. 1. Introduction Since the early works from Willert and Gharib (1991), Digital PIV (Particle image Velocimetry) is recognized as a relevant velocimetry technique. Nevertheless, the complexity and variety of the technique has precluded an appropriate assessment of its measurement confidence interval, up to present time. This assessment is necessary to allow proper comparison between experimental results and numerical calculations. In turn, this comparison is one of the engines in the development of key devices for industries. Given this scenario, increasing insight into the different errors has been a continuous subject of research (Charonko JJ and Vlachos 013, Sciacchitano et al. 013, Wilson and Smith 013, Kähler et al. 01, Legrand et al. 01, Timmins et al. 01, Nobach 011, Nogueira et al. 011, Westerweel 1998, among many others). These stepping stones have generated the possibility to shift from basic and particular knowledge towards proper error handling and assessment procedures. For a correct description of the error, the confidence interval has to be accompanied with additional information. Some flow quantities like dissipation and turbulent kinetic energy (tke), where the contribution of the small scales may be relevant, require the assessment of which is the smallest scale resolved in the measurement. This is important because it is common to have such a broad range of different spatial scales in a fluid flow that it is impossible to accurately measure all the different scales simultaneously. In such situations, a flow quantity (like tke) contained in a certain range of scales is different to the one contained in another different range. As only the error of what has been measured can be assessed, the values and uncertainty intervals do not necessarily match. Figure 1 depicts this scenario. The uncertainty interval would not have any sense if the range of measured scales is not specified. The consequence is that the assessment of the smallest measured scale and its error is customary when dealing with the accuracy of the measurement. Furthermore, the assessment for these scales has to take into account that part of their errors may come from different scales in the flow. The resulting confidence interval has to include all the error for the measured scales (including the one injected in these scales from other scales in the flow). Of course, it cannot include the amount of flow magnitude that is contained in the scales that have not been correctly measured. The magnitude of the - 1 -

2 Lisbon, Portugal, July, 014 contribution from not resolved scales has to be obtained in a different measurement, or calculated theoretically based on the knowledge of the physics of the flow. One example is to resort to reasoning on the turbulent cascade and the size of the Kolmogorov scale in relation to the smallest measured scale. Also for this, the assessment of which is the smallest measured scale and its error is crucial. 1.80E E E-0 1.0E E E E E-03.00E E r/d r/d Fig. 1 Non dimensional turbulent kinetic energy (tke) measured by means of PIV at the outlet of a jet engine nozzle of diameter D (Nogueira et al 009). a) tke contained in scales down to 0.04D. b) tke contained in scales down to 0.01D. This paper starts by considering, in Section, which are the constraints that limit the smaller measureable scales. Although the subject has been already commented in a previous work by the authors (Nogueira et al. 01), for the sake of clarity, a brief consideration of this issue seems customary. Once this base is established, tools to assess the smallest scale resolved by the PIV measurement are proposed in Sections 3, 4 and 5. This is done by first introducing the methodology and then obtaining results on synthetic images. One of the tools proposed for the study is a classical structure function of the flow. The results to assess the smallest measured scale in a PIV measurement are promising. The way to use it for such assessment is also detailed in Section 5, complemented with the study of the coherence between the results from synthetic and real images in Section 6.. Range of available spatial scales in a PIV measurement At the origins of the Digital PIV instrument and for most cases it could be considered that the range of spatial scales available in a measurement was defined only by two conditions: (i) The largest scale was limited by the size of the flow field extension imaged by the CCD sensor. (ii) The smallest scale (measured in pixels) was defined by the algorithm capacity to resolve small scales. The available scales range at different conditions was determined by the shadowed zone in Figure a. It was only a matter of selecting the right optical magnification to set an experiment. In this paradigm it was given for granted that the laser sheet thickness was small enough to properly adquire the smallest scales that the PIV algorithm was able to resolve. As the sensors grow in number of pixels and the algorithms in capacity to resolve smaller scales, an additional boundary may come into play and the laser sheet thickness becomes a limiting factor. In fact, the laser sheet thickness cannot be arbitrarily reduced. It is coupled with two additional constraints that block the possibilities of such reduction. All together, the three coupled constraints limit the available scales through the following rationale: (i) The laser sheet thickness, d, has to be significantly smaller than the smallest spatial scale to resolve, l L. Otherwise, the variations across the laser sheet would overlap from the imaging sensor point of view, precluding a correct measurement. Considering a small factor, f 1 < 1, to define the allowable size, we arrive to the following expression: -.00E

3 Lisbon, Portugal, July, 014 (ii) d f 1 l L (1) The laser sheet thickness, d, has to be large enough so that the loss of particle pairs in PIV due to the largest out of plane velocity U T is a small percentage, f < 1: U T t f d () (iii) The presence of the time between laser pulses, t, at Equation indicates that its reduction would allow reducing the laser sheet width. But in turn this reduction incorporates new limitations. In particular, t has to be large enough to assure that the sub-pixel resolution accuracy, sr, is significantly smaller than the distance traveled by the smallest velocity to measure, u L. Considering a small factor, f 3 < 1, to define the allowable ratio, we arrive to: f 3 u L t sr (3) Combining these three constraints, it can be concluded that the following expression applies as a necessary condition: U T /u L f 1 f f 3 l L /sr (4) In addition, any flow incorporates a link between U T /u L and their corresponding scales l T /l L. For example, in homogeneous isotropic turbulence l T /l L = (U T /u L ) 3 in the inertial range, resulting in: l T /l L (f 1 f f 3 l L /sr) 3 (5) For other flows and ranges the expression may somehow differ, but in general we arrive to a situation like the one depicted in Figure b. Here, the largest range of available scales is obtained for a certain magnification with different limitations than the one obtained from Figure a. In this condition, the limit for the smallest measured scale comes from the coupling between laser sheet thickness, d, and time between laser pulses, t. This coupling is contained in Equation 5, with the rationale expressed above. It can be observed that under these conditions, the algorithm capacity to resolve small scales is not a limiting factor any more for the largest range of available spatial scales. Largest range of scales available Largest range of scales available Fig. Limits for the available scales range. a) Case where the thickness of the laser sheet does not generate additional conditions. b) Case where the limitations from the coupling between d and t come into play. Further discussion about these concepts for different Reynolds numbers can be consulted at Nogueira et al. (01) but it is out of the scope of this paper. A consequence of the concepts commented in this section is that assessing the error related to the laser sheet thickness is a relevant issue. The sections below propose a tool to assess this error in real images, as well as the sensitivity for different small scales in the PIV measurement and for changes in the laser sheet thickness

4 Lisbon, Portugal, July, Assessment of the smallest scale resolved in a PIV measurement Errors related to the sub-pixel resolution can be assessed by means of multiple t strategies, available to any conventional PIV setup (Nogueira et al. 011). This is so because the error magnitude does scale with the pixel size and not with the time between laser pulses. This allows discriminating them from the measured velocities, which do scale with the time between laser pulses. Occurrence of outliers, due to excessive loss of particle pairs because of out of plane velocity, is also apparent by visual inspection of the flow field. Usual post-processing filters like the median one (Westerweel 1994) give information on their importance and number. The case is different for the errors related to the algorithm capacity to resolve small scales and the effect of the small scales contained across the laser sheet thickness. A priori both errors do scale with the time between pulses making it difficult to discriminate them from the measured velocity. In addition, both affect the small scales, making it also difficult to discriminate one from the other. Limits on the capacity of the PIV algorithm to resolve small scales are usually known to a certain degree. For conventional PIV algorithms they are established since the very beginning of Digital PIV (Willert and Gharib, 1991). The purpose of this paper is to propose tools and procedures for assessing the magnitude of the error related to the laser sheet thickness at small scales and to discriminate it from the one associated to the PIV algorithm spatial resolution. In order to assess which is the smallest scale that has been correctly resolved in a PIV measurement and the source of error that precludes solving smaller ones, a tool to discriminate scales has to be selected. For this work, promising results have been obtained using a magnitude derived from the flow structure functions defined in the classical studies on turbulence (Pope 000). In particular, the characteristic velocity difference, u l, over a given distance, l, can be defined as: u l = [U(x+ l)-u(x)] (6) The angular brackets operator,, means average over the measured flow field. Equation 6 is actually a general definition for several second order structure functions. In this work the following two are considered: u x lx = [U x (x+ lx)-u x (x)] (longitudinal structure function) (7) u x lz = [U x (x+ lz)-u x (x)] (transversal structure function) (8) The sub-indices introduced indicate the direction for the velocity and for the evaluated distance. Sub-index z corresponds to the direction perpendicular to the laser sheet and sub-index x corresponds to a direction contained in the laser sheet. The possibility to calculate other axis combinations results in additional data, but the rationale behind would be the same that the one explained in this paper for these two ones. Only values of these functions for small scales are needed and presented in the paper. This is an additional advantage as the structure function has a fast statistical convergence for small values, requiring a low number of PIV snapshots. Focusing on the small scale error assessment, the expected effects on u x lx differ for the two commented error sources for small scales. This allows discriminating errors coming from each source. The rationale and characteristics of this difference is as follows: (i) The spatial resolution of the algorithm, for conventional multi-grid PIV (Soria 1996) is related to the interrogation window acting in a similar way as a moving average i.e. a low-pass filter. In general the rest of the algorithms act in a similar way, even when the interrogation window is not the limiting factor (Scarano 00, Astarita 009, Nogueira et al. 001, among others). For u x lx the characteristic effect of a moving average is a relative growing attenuation for small scales. This effect is schematically represented in Figure 3a. In addition to that, a constant difference between the real u and the measured u x lx x lx is expected for large scales. This is so because u x lx can - 4 -

5 Lisbon, Portugal, July, 014 (ii) be considered as the addition of the mean square value of the velocity differences at large scales (not attenuated) plus the mean square value of the velocity differences at small scales (attenuated). This can be assumed if both ensembles of velocity differences have zero average and are statistically independent between them. Two different ranges of spatial scales will be considered for the case of the laser sheet thickness, to explain its effect on the measurement of u x lx. From the largest scales down to the scale of the laser sheet width, the effect is similar to the one in the previous case. It acts as a low-pass filter due to a moving average effect across the laser sheet. Large scales are not attenuated and a constant difference between the real u and the measured u x lx x lx is expected (as resulted in the previous section, due to the composition between large and small scales in u x lx for large values of lx). Approaching the scale of the laser sheet width, an increasing attenuation is expected in coherence with its low-pass nature. But the effect changes radically for scales smaller than the laser sheet width. For these scales spurious contributions of large amplitude are expected. This is represented in Figure 3b for u x lx. The reason for these spurious contributions is that flow transversal contributions to u x lz for lz in the order of the laser sheet thickness can be considered by the PIV system as contributions to u x lx for erroneously much smaller measured values of lx. This behavior is depicted in Figure 4. As generally u l diminishes for smaller l, this effect spuriously increases the value of the measured u x lx for these scales, smaller than the laser sheet width. Fig. 3 Expected effect of small scale error sources on u x lx. a) Effect from the spatial resolution of the algorithm. b) Effect from the laser sheet thickness. Fig. 4 Mechanism that includes relatively large values from u x lz as spurious contributions to u x lx, for d > lz >> lx. It can be observed that the view from the CCD may associate a relative distance between particles much smaller than the real one

6 Lisbon, Portugal, July, 014 Based on the different behavior of the measured u x lx for the different sources of small scale errors, depicted in Figure 3, this work proposes using it as a tool for assessment of the small scales error. 4. Methodology This section describes the methodology followed to check the suitability for error assessment of the hypothesized behavior in Figure 3. The behavior of u x lx is checked here using synthetic images. This practice is common in PIV. Measuring a synthetic image allows direct comparison between the real displacement of the particles and the measured one. Additionally, the flow can be modified to check particular issues. For example, setting the out of plane velocity component V z to zero would eliminate errors coming from the out of plane loss of particles. Alternatively setting to zero the derivatives in respect to the z direction would eliminate errors coming from variations across the laser plane width. Once the error assessment possibilities have been tested on synthetic images, u x lx is calculated using real ones. Coherence between the results using synthetic and real images is checked to finalize the suitability evaluation. The flow field selected for synthetic images corresponds to the data of a DNS (Direct Numerical Simulation) of homogeneous isotropic turbulence. Its database is publically available (Li et al. 008). Taylor Re number is 433. The synthetic image generator produces 4k by 4k images where the Kolmogorov scale corresponds to 1.9 pixels. As the purpose of these synthetic images is to test the ability to cope with difficulties arising from the scales in the flow, they contain no noise except for the spatial discretisation of the simulated image sensor and an effective 16-bit grey level sampling. The average distance between the randomly located particle images is, δ = 5 pixels, i.e. 4/(π δ ) 0,05 ppp (particles per pixel). The e - diameter of all the Gaussian particle images is d p =. pixels. The Gaussian shape of the particle images is integrated with unity fill factor over each square pixel surface (Westerweel 1998). A Gaussian intensity profile was selected for the light sheet, defining the laser sheet thickness by the location of the e - values in respect to the central intensity. Where particles overlap, the corresponding intensities were added. For all cases (real and synthetic images) the PIV vector fields have been computed using commercial software (DaVis 7.0 by LaVision). The analysis consisted in a multi-grid scheme with image correction using Whittaker interpolation for grey levels and ending with pixels interrogation windows at 75% overlapping. 5. Results from synthetic images Two cases are presented in this section. For the first one, only the DNS velocity values at the center of the laser sheet are used. Variations across the plane are eliminated (d()/dz = 0) and the thickness of the laser sheet is selected so that the out of plane velocity is negligible (less than 3% of particle pair loss due to out of plane velocity). Under these conditions the laser sheet thickness does not introduce any significant error. The time interval corresponds to DNS time units. The results clearly match what was expected from Section 3. For large scales difference (u x lx ) DNS - (u x lx ) PIV is constant. For small scales (u x lx ) PIV is attenuated so the response of (u x lx ) DNS / (u x lx ) PIV diminishes for smaller frequencies. This result is plotted at Figure 5a

7 Lisbon, Portugal, July, 014 Fig. 5 Test on the expected effect of small scale error sources on u x lx. a) Effect from the spatial resolution of the algorithm. b) Mixed effect from the spatial resolutions of the algorithm and from the laser sheet thickness, d = 100 pixels, (mixing of effects from Figures 3a and 3b). The second case consisted in retaining all the DNS variations across the laser sheet. Difference in error between the previous case and this can be attributed to the lasers sheet thickness. Again the results match what was expected from Section 3. As the low-pass effect of the PIV algorithm cannot be avoided, both effects from Figures 3a and 3b are mixed here. For large scales the difference (u x lx ) DNS - (u x lx ) PIV is constant. As the scales approach the thickness of the laser sheet, (u x lx ) PIV approaches and crosses (u x lx ) DNS towards larger values generated from the mechanism explained in Figure 4. For scales smaller than the spatial resolution of the PIV algorithm (u x lx ) PIV diminishes due to the low-pass effect described in Figure 3a. This result, combination of the described effects, is plotted in Figure 5b. 6. Results from real images The objective of the work here reported is its application on real images. Even though the detailed flow field is unknown, thus impeding to offer the error, the tests in this section are customary for checking the coherence between the results found with synthetic images and those corresponding to real ones. The PIV system used to acquire the real images consists of a double cavity 380 mj per pulse Nd-Yag laser from QUANTEL and a LaVision FlowSense CCD camera of k by k pixels covering an area of about 131 by 131 mm. The magnification corresponds to mm/pixel. The flow was seeded with food-grade glycol droplets with diameter in the order of 1 µm generated by means of Laskin Nozzles (Kähler et al. 00). Time between laser pulses was t = 100 µs. The flow used for this test corresponds to a meridian plane of a cold model of a swirl stabilized cylindrical burner. No swirl has been introduced to reduce out of plane velocity. The measurement has been performed 35 cm downstream of the nozzle. The nozzle geometry is complex, being out of the scope to detail it, as the measurements objective is just to check the procedures to assess small scale errors related to the laser sheet thickness. The relevant issue is that the estimated Kolmogorov scale corresponds to 0.6 pixels and the Re number based on the nozzle exit diameter exceeds A set of lenses on a telescope-like mounting was used to change the effective diameter of the laser beam before the laser sheet formation. Changing its value was easily done during experimental runs just by rotating the mounting. The laser sheet width was measured for the different positions of the lenses, so the laser sheet width was known from the set of lenses positions. Measurement results for u x lx at d = 1.5 mm and d = 3 mm are presented in Figure 6. The rationale behind it is that in the results of real images (u ) x lx REAL is not available and coherence will be checked through comparing (u ) x lx PIV at two different laser sheet thicknesses

8 Lisbon, Portugal, July, 014 Fig. 6 Comparison of u x lx.measured for two different laser sheet widths. a) Real images (brown line for d = 3 mm = 47 pixels and green line for d = 1,5 mm = 3 pixels). b) Synthetic images result for comparison (brown line for d = 100 pixels and green line for d = 50 pixels) Differences are evident for Figures 6a and 6b, indicating that the effects coming from the size of the Kolmogorov scale, the seeding density, particle size and other issues have to be studied. Nevertheless, the crossing of the lines leaving the one corresponding to the larger laser sheet thickness on top of it, clearly indicates the presence of the mechanism indicated in Figure 4. In addition, the difference between both lines in Figure 6a indicates that the magnitude of this error for laser sheet thickness of 3 mm is not negligible. Further work is advisable, but the role of structure functions like u x lx for small scales error assessment seems promising. 7. Conclusions Aiming at measuring a large range of different spatial scales simultaneously, sensors with a large number of pixels are required. For them, the experiment setup optimization may lead to conditions in which the laser sheet width should have a prescribed value that makes it the limiting factor for small scales. Reducing the laser sheet width in these conditions reduces the range of different measurable scales instead of increasing it, due to coupling between laser sheet width and time between laser pulses. Within this scenario, error assessment for small scales is customary. The second order structure function of the flow is an intuitive and easy to implement tool to quantify characteristic velocity differences at a particular scale and thus to reach insight into the effects from the laser sheet thickness for small scales. In addition, variation of the laser sheet thickness is not excessively complex. The combined use of these two concepts seems promising in respect to develop a procedure for the assessment of errors due to the laser sheet width. Using these concepts in this paper has allowed verifying that the variations of the laser sheet thickness generate spuriously large values for the velocity associated to small scales. This paper also shows that the magnitude of the error coming from this source is not negligible. Further work is advisable for the full characterization of this tool and its participation in error handling procedures. 8. References Astarita T (009) Adaptive space resolution for PIV. Experiments in Fluids, Vol.: 46/6, pp: Charonko JJ and Vlachos PP (013) Estimation of uncertainty bounds for individual PIV measurements from cross correlation peak-ratio. Meas. Sci. Technol. Submitted Kähler CJ, Scharnowski S, Cierpka C. (01) On the resolution limit of digital particle image - 8 -

9 Lisbon, Portugal, July, 014 velocimetry. Experiments in Fluids. 5/6: Kähler CJ, Sammler B, Kompenhans J (00) Generation and control of tracer particles for optical flow investigations in air. Experiments in Fluids. 33: Legrand M, Nogueira J, Ventas R, Lecuona A (01) Simultaneous assessment of peak-locking and CCD readout errors through a multiple Delta t strategy. Experiments in Fluids. 53/1: Li Y, Perlman E, Wan M, Yang Y, Meneveau C, Burns R, Chen S, Szalay A and Eyink G (008) A public database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. Journal of Turbulence 9/31: Nobach H (011) Influence of individual variations of particle image intensities on high-resolution PIV. Experiments in Fluids. 50/4: Nogueira J, Lecuona A, Ruiz-Rivas U, Rodríguez PA (001) Analysis and alternatives in twodimensional multigrid particle image velocimetry methods: application of a dedicated weighting function and symmetric direct correlation. Meas. Sci. Technol. 13: Nogueira J, Lecuona A, Nauri S, Legrand M & Rodríguez PA (009) Multiple t strategy for particle image velocimetry (PIV) error correction, applied to a hot propulsive jet. Meas. Sci. Technol. 0 Nogueira J, Lecuona A, Nauri S, Legrand M, Rodriguez PA (011) Quantitative evaluation of PIV peak locking through a multiple Delta t strategy: relevance of the rms component. Experiments in Fluids. 51/3: Nogueira J, Lecuona A, Legrand M, Rodriguez PA, Ventas R and Rodríguez-Hidalgo MC (01) Setup optimization for PIV measurements in presence of turbulence. 16th. Int. Symp. on Applications of Laser Techniques to Fluid Mechanics. Lisbon, Portugal, 09-1 July, 01. Pope SB, 000, Turbulent flows Ed. Cambridge University press. Soria J (1996) An Investigation of the near wake of a circular cylinder using a video-based digital cross-correlation particle image velocimetry technique. Exp. Thermal Fluid Sci. 1: 1 33 Scarano F (00) Iterative image deformation methods in PIV Meas. Sci. Technol. 13: R1 19 Sciacchitano A, Wieneke B and Scarano F (013) PIV uncertainty quantification by image matching. Meas. Sci. and Technol. 4: Timmins BH, Wilson BW, Smith BL and Vlachos PP (01) A method for automatic estimation of instantaneous local uncertainty in particle image velocimetry measurements Exp Fluids. 53: Westerweel J (1994) Efficient detection of spurious vectors in particle image velocimetry data, Experiments in Fluids. 16: Westerweel J (1998) Effect of sensor geometry on the performance of PIV. 9th Int. Symp. on Applications of Laser Techniques to Fluid Mechanics. Instituto Superior Técnico, Lisbon, Portugal. Willert C and Gharib M (1991) Digital Particle Image Velocimetry. Exp. Fluids 10-4: Wilson BM and Smith BL (013) Uncertainty on PIV mean and fluctuating velocity due to bias and random errors. Meas. Sci. Technol. 4: Acnowledgements This work has been partially funded by the Spanish Research Agency grants ENE TERMOPIV: Combustión y transferencia de calor analizadas con PIV avanzado ; and ENE TERMOPIV II ; the regional grant CCG08-UC3M/ENE-443 PIVEROT: Mejora en la medida PIV de campos fluidos de quemadores LSB por reducción de errores producidos por efectos termopsicrométricos en los sensores CCD ; and the Madrid Community grant CCG10-UC3M/ENE-516 ES-COMB: Estructuras coherentes en quemadores de combustión limpia. We would like also to express a special acknowledgement to the laboratory technicians Mr. Manuel Santos, Mr. Carlos Cobos, Mr. Israel Pina and Mr. David Díaz for their appreciated help in the experimental setup elaboration