geosciences Article
Efficiency of a Digital Particle Image Velocimetry (DPIV) Method for Monitoring the Surface Velocity of Hyper-Concentrated Flows Donatella Termini *
and Alice Di Leonardo
Department of Civil, Enviromental, Aerospace, Materials Engineering (DICAM), Viale delle Scienze, 90128 Palermo, Italy;
[email protected] * Correspondence:
[email protected] Received: 9 August 2018; Accepted: 16 October 2018; Published: 19 October 2018
Abstract: Digital particle image velocimetry records high resolution images and allows the identification of the position of points in different time instants. This paper explores the efficiency of the digital image-technique for remote monitoring of surface velocity and discharge measurement in hyper-concentrated flow by the way of laboratory experiment. One of the challenges in the application of the image-technique is the evaluation of the error in estimating surface velocity. The error quantification is complex because it depends on many factors characterizing either the experimental conditions or/and the processing algorithm. In the present work, attention is devoted to the estimation error due either to the acquisition time or to the size of the sub-images (interrogation areas) to be correlated. The analysis is conducted with the aid of data collected in a scale laboratory flume constructed at the Hydraulic laboratory of the Department of Civil, Environmental, Aerospace and of Materials Engineering (DICAM)—University of Palermo (Italy) and the image processing is carried out by the help of the PivLab algorithm in Matlab. The obtained results confirm that the number of frames used in processing procedure strongly affects the values of surface velocity; the estimation error decreases as the number of frames increases. The size of the interrogation area also exerts an important role in the flow velocity estimation. For the examined case, a reduction of the size of the interrogation area of one half compared to its original size has allowed us to obtain low values of the velocity estimation error. Results also demonstrate the ability of the digital image-technique to estimate the discharge at given cross-sections. The values of the discharge estimated by applying the digital image-technique downstream of the inflow sections by using the aforementioned size of the interrogation area compares well with those measured. Keywords: digital particle image technique; surface velocity; remote monitoring; experiments
1. Introduction Accurate knowledge of flow discharge is of crucial importance especially for hydrologist and fluvial geomorphologists. In a context of a changing climate, significant hazards associated with both floods and debris or mud flows occur more and more frequently, causing significant changes in topography and river morphology [1]. Traditional techniques usually are based on current-meter measurements (from bridges or set across the channel width) and on the linking of water depth with the discharge through the rating flow velocity curve. Due to the limited accuracy of the current-meter, more modern and accurate instruments, such as Acoustic Doppler Velocimeters (ADV), are also used to measure distribution [2,3]. But, the use of the aforementioned measurement techniques presents many practical difficulties and requires long acquisition times. In particular, Acoustic Doppler Velocimeters require water depths
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that limit their use in shallow water flows. For this reason, these techniques are essentially limited to low flow conditions inducing high uncertainty in rating curve extrapolation for higher stages. In fact, high flow conditions, characterized by high velocities, large water depths, and high sediment concentrations do not permit the use of the traditional measurement equipment and are not safe for operators because flow depth can significantly change and high flow velocities and floating debris occur [3]. Thus, in recent years, many efforts have been made both to develop techniques capable of providing reliable velocity measurements under challenging conditions and to identify alternative methods to estimate water discharge. As an example, some researchers [4–6], based on the application of the entropic concept, suggested to evaluate the mean flow velocity, and de facto the discharge, through the information of the surface velocity, with the great benefit of having quick measurement and high safety for the person involved in measurement. Other researchers have suggested to associate the information of surface velocity with the water level information, leading the estimation either of the depth-averaged velocity value by applying the dimensionless logarithm law [7] or of the water discharge [8]. In such a context, in the last decade, because of improvement of computing capabilities, image-based techniques have been increasingly used for surface velocity measurements in field [7–10] and in laboratory channels [11–14]. The advantages of image-based techniques compared to traditional techniques are essentially related to the fact that they are non-intrusive and allow obtaining simultaneous spatial information about the instantaneous velocity components also in unsteady flows [7]. Thus, it is possible to yield a large amount of data in a rather short measuring time and to calculate simultaneously the average values for identical spatial windows over many images. Consequently, quantitative measurement of fluid velocity vectors at a very large number of points could be obtained simultaneously and with a very low cost [15–17]. In field applications, large scale PIV (LSPIV) technique has been used to measure the surface flow velocities especially in complex situations such as at junctions between channels or around hydraulic structures [8–18]. In laboratory channels, the conventional PIV technique has been especially used to estimate surface velocity in very small water depths which make the use of direct measurements with traditional equipment impractical [19]. There are some differences between field and laboratory applications of imaging technique [20]. Large scale PIV (LSPIV) covers larger field of view and uses more inexpensive (even natural) illumination devices than conventional PIV [18]. Furthermore, if the flow is dense enough, the objects carried by it could be used as traceable patterns so that artificial seeding is required only when few natural tracers are visible at the surface of the flow. In this case, the tracers need to be larger than the resolution of one pixel to be followed, while in laboratory PIV applications, flow is seeded with microparticles. Thus, recently, image techniques have been also used for hyper-concentrated flows also in laboratory channels [21,22]. The high density of the natural grains allows an easier application of the PIV procedure with natural tracers but, since the granular material is non-transparent, the measurements inside the flow are generally impossible so that the laser sheets are inadequate to illuminate the particles and alternative lighting systems, such as halogen lamps, have to be used [22]. Pudasaini et al. [23,24] employed flashes, Sheng et al. [25] used halogen lamps powered by a flickering-free ballast, Sarno et al. [26–28] employed a high-brightness no-flicker LED lamp. The application of image-based techniques in hyper-concentrated flows is still matter of debate, especially in relation to the opportunity to use moving grains as natural tracers and to the identification of the acquisition time determining adequate number of images to be processed. From the aforementioned, it is clear that, especially in LSPIV applications, appropriate analyses and processing procedures have to be used in order to obtain quantitative information by applying image-based techniques [22,29]. One of the challenges in the application of image-techniques is the determination of the error in estimating surface velocity. The error quantification is complex and it
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depends on many factors characterizing either the experimental conditions and/or the processing algorithm used. Because of the practical difficulties and the complexity of the required equipment in field applications, performing investigations in a controlled environment, such as the laboratory, is often preferred. Many works have been conducted in order to analyze errors arising from the experimental conditions and measured flow. Very few works have focused attention on errors caused by the processing method. Huang et al. [30] highlighted that two major errors could arise in processing procedure. One error could be related to the quality of images and is due to the background noise (always present in images). This error is a random function of time and space and causes differences between the two patterns; this error is generally corrected (or reduced) by an image pre-processing procedure [31]. Another error could occur from insufficient data (see also in [26]) which could be due either to a lack of flow tracers or to poor quality images or, since velocity measurements are performed in a specific range, to the acquisition frequency and/or the acquisition time which determine the number of images to correlate. Another cause of error could be related to the size of the particle imaging pattern matching (or of the interrogation area) in a pair of images to correlate. This error occurs when particles imaged in the first image are not found in the second image (the so-called “out of-pattern-motions”—see [31,32]). Thus, the accuracy of the results could be affected either by the number of frames to be processed or by the dimension of the interrogation area. It is quite difficult to isolate which parameter is more restrictive for the accuracy of the imaging technique. In the present paper, the aforementioned parameters (i.e., the number of frames to be processed and the dimension of the interrogation area) are considered and the efficiency of the digital image-technique for remote monitoring of surface velocity and discharge in hyper-concentrated flow is studied. The analysis is conducted in a scale flume with a complete installation to record a high number of images with a high-precision camera. The novelty of the presented analysis is related to the fact that the experiments have been conducted under controlled laboratory conditions but with images acquisition conditions similar to those generally adopted in the field [9,22]. This allows us to analyze the efficiency of the procedure varying the flow characteristics and the sediment concentrations more easily than in the field. In previous works, the recorded images were used to analyze the influence of the geometrical parameters on the hyper-concentrated flow propagation [33], to explore the utility of the PIV technique and to compare the accuracy of results obtained by using artificial tracers with those obtained by using natural tracers conveyed by the stream [34,35]. In the present paper, the PIV technique is applied (by the help of PivLab code 1.32 in Matlab [36–38]) by considering natural floating tracers and the specific aims are twofold: (1) to investigate the error in free surface velocity estimation with respect to the number of frames to be processed and to the dimension of the interrogation area; (2) to verify the applicability of the procedure to evaluate the flow discharge through the comparison with discharge estimates by using an ultrasonic instrument. The experimental apparatus and the estimation procedure are described in Section 2; the sensitivity analysis and the discharge estimation are presented in Section 3. Finally, conclusions are drawn in Section 4. 2. Materials and Methods 2.1. Experimental Apparatus and Conditions An experimental program has been recently conducted at the Hydraulic laboratory of the Department of Civil, Environmental, Aerospace and of Materials Engineering (DICAM)—University of Palermo (Italy) in order to analyze the propagation phenomenon of hyper-concentrated flow in a defense channel. A series of experiments was conducted in a 1:35 scale flume reproducing the defence channel under construction in a small urban area of Messina’s territory (Italy). Such a defence channel was designed after a huge rainfall in 2009 that mobilized large amount of sediments causing damages and death. The experimental apparatus is shown in Figure 1. As Figure 1 shows, the main channel
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includes three denominated respectively as I1, I2, and I3. The bed is concrete and Geosciences 2018, inflow 8, x FOR channels PEER REVIEW 4 of 17 the channel’s walls are Plexiglas strips. The channel’s width is equal to 23 cm in the reach between section and section 15Figure (see Figure 1c) it and it is equal 17 cm from section 18 until section 1 and 1section 15 (see 1c) and is equal to 17tocm from section 18 until thethe lastlast section of of the the channel. channel.
(a)
(b)
(c)
Figure Experimentalapparatus: apparatus:(a)(a)Render; Render;(b) (b)Plane-view Plane-viewofofthe the images covering the examined Figure 1. 1. Experimental images covering the examined channel; (c)(c) Examined transversal sections. channel; Examined transversal sections.
During the experiments, thethe water–sediment mixture waswas recycled through a pump located in thein During the experiments, water–sediment mixture recycled through a pump located downstream reservoir. The water–sediment mixture was obtained by using field sediments (medium the downstream reservoir. The water–sediment mixture was obtained by using field sediments sediment = 2 mm)d50and includes sediment diameter between (mediumdiameter sedimentd50diameter = 2itmm) and it includesparticles sedimentwith particles withranging diameter ranging −1 mm and−12 × 10−3 mm. The −3 1× 10 sediment concentration varied in the range 10%–50%. During the between 1 × 10 mm and 2 × 10 mm. The sediment concentration varied in the range 10%–50%. experiments, the inflow discharge was measured both upstream of the inflow channels and in peculiar During the experiments, the inflow discharge was measured both upstream of the inflow channels sections (sectionssections 4, 5, 11,(sections 12 of Figure 1c)12 along the channel bythe using the ultrasonic instrument and in peculiar 4, 5, 11, of Figure 1c) along channel by using the ultrasonic Mainstream EH7000 (by Endress + Hauser S.p.a.). In the present work, attention is focused onis instrument Mainstream EH7000 (by Endress + Hauser S.p.a.). In the present work, attention
focused on experimental runs conducted with sediment concentration of around 50% and values of the discharge at inflow channels respectively equal to QI1 = 0.004 m3/s, QI2 = 0.00135 m3/s, QI3 = 0.00174 m3/s. 2.2. Acquisition and Processing Methodology
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experimental runs conducted with sediment concentration of around 50% and values of the discharge at inflow channels respectively equal to QI1 = 0.004 m3 /s, QI2 = 0.00135 m3 /s, QI3 = 0.00174 m3 /s. Geosciences 2018, 8, and x FOR PEER REVIEW 2.2. Acquisition Processing Methodology
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The (1) recording recording images; pre-processing, which The PIV PIV technique technique consists consists of of four four steps: steps: (1) images; (2) (2) pre-processing, which allows allows the improvement of the information to be taken from the images; (3) evaluation, which the improvement of the information to be taken from the images; (3) evaluation, which allows allows the the application of the the cross-correlation cross-correlationalgorithm; algorithm;(4) (4)post-processing, post-processing,which which allows interpretation application of allows thethe interpretation of of results. The processing procedure was carried out by the help of the PivLab algorithm in Matlab. results. The processing procedure was carried out by the help of the PivLab algorithm in Matlab. A A high speed camera (AOSTechnology TechnologyAG) AG)was wasused usedtotorecord recordthe theflow flowmotion. motion. In In order order to to reproduce reproduce high speed camera (AOS conditions similar to those in field, the images were acquired either by natural lighting or conditions similar to those in field, the images were acquired either by natural lighting or by by using using halogen lamps opportunely positioned to obtain a homogenous illumination [35,39]. Preliminary runs halogen lamps opportunely positioned to obtain a homogenous illumination [35,39]. Preliminary were in order set illumination conditions and to select rate ofthe therate acquisition frequency runs conducted were conducted in to order to set illumination conditions and the to select of the acquisition as function of the camera’s characteristics and the number of images to cover the channel. During the frequency as function of the camera’s characteristics and the number of images to cover the channel. preliminary runs, the streamwise velocity component was also measured at some control points During the preliminary runs, the streamwise velocity component was also measured at some control opportunely selected along the channel by using acoustic velocityvelocity profilerprofiler (DOP2000 by signal points opportunely selected along the channel by an using an acoustic (DOP2000 by processing s.r.l.). Finally, the illumination system consisted of 7 halogen lamps distributed along the signal processing s.r.l.). Finally, the illumination system consisted of 7 halogen lamps distributed boundaries of the channel’s and positioned the channel adequate so as toheight have along the boundaries of thearea channel’s area and above positioned above at theanchannel at height an adequate asohomogeneous and diffused illumination on the surface without shadows or light reflections [40]; as to have a homogeneous and diffused illumination on the surface without shadows or light another halogen lamp (type ARRI 2000W with filter lens and electric ballast) centered upon the channel reflections [40]; another halogen lamp (type ARRI 2000W with filter lens and electric ballast) centered and on and the top of the camera also (seewas Figure uponcollocated the channel collocated on the was top of theused camera also2).used (see Figure 2).
(a)
(b) Figure 2. (a) Scheme of the halogen lamps’ locations; (b) Scheme of images acquisition system.
As described describedinindetail detail [38], a rate offrames 600 frames per second withresolution image resolution of in in [38], a rate of 600 per second with image of 640 × 240 640 × 240 pixels (px) was chosen and 22 images were considered to cover the examined channel pixels (px) was chosen and 22 images were considered to cover the examined channel reach (see reach 1b).to Inmaintain order to maintain a constant optical cone the width, the camera was placed on a Figure(see 1b).Figure In order a constant optical cone width, camera was placed on a plane plane parallel the channel’s bedcentered and centered the channel at a constant distance ofm 1.5from m from parallel to theto channel’s bed and uponupon the channel at a constant distance of 1.5 the the bed. Furthermore, both target points and reference meters were positioned on the bed and on the bed. Furthermore, both target points and reference meters were positioned on the bed and channel’s correction of the The target pointspoints and the reference meters channel’s walls wallsfor forthe thegeometrical geometrical correction of images. the images. The target and the reference were used to used applytothe calibration and rectification toolboxtoolbox for MATLAB and to calibrate the length meters were apply the calibration and rectification for MATLAB and to calibrate the ∼ scale the images. It was estimated that 1 px = 11.2 lengthofscale of the images. It was estimated that pxmm. ≅ 1.2 mm. During the experimental run, the particles of the water–sediment mixture were considered as natural tracers for the PIV analysis. For each image, a sequence of frames was recorded by means of the high-speed camera. The The number number of of recorded recorded frames frames per per image image ranged ranged between between 4800 4800 and and 6000. 6000. Then, and floating wooden particles of 10 in diameter and Then, artificial artificialtracers tracersconsisting consistingofofinsoluble insoluble and floating wooden particles ofmm 10 mm in diameter 3and mm high were released into the hyper-concentrated flow. The artificial particles were painted with 3 mm high were released into the hyper-concentrated flow. The artificial particles were painted yellow phosphorescent color incolor orderintoorder ensuretotheir visibility. Once theseOnce particles released in with yellow phosphorescent ensure their visibility. thesewere particles were released in the hyper-concentrated flow, sequences of images were again recorded at constant time intervals by means of the high-speed camera. This allowed us to track the particles trajectories at different times. The information obtained by using the artificial seeding were used in the preprocessing step in order to eliminate some errors before correlation process, to remove peculiar anomalies from the considered dataset [32] and to calibrate the pre-processing camera settings
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the hyper-concentrated flow, sequences of images were again recorded at constant time intervals by means of the high-speed camera. This allowed us to track the particles trajectories at different times. The information obtained by using the artificial seeding were used in the pre-processing step in order to eliminate some errors before correlation process, to remove peculiar anomalies from the considered dataset [32] and to calibrate the pre-processing camera settings [34,35]. For the calibration, the values of flow velocity components previously measured by using the acoustic velocity profiler during the preliminary runs were also used. A high-pass filter was applied to remove the background signal [36] and the FFT window deformation was considered. For each image, the region of interest (ROI) was identified by excluding the pixels external to the flow field domain. To apply the PIV method, each frame was thought as an independent sample of the velocity field and the statistical sampling theory was applied. According to previous works [7,31], the most probable displacement of a group of particles travelling on a straight line between two consecutive images was determined by applying the spatial cross-correlation method [41]. Each pair of frames was cross-correlated in order to obtain the most probable particles displacement in small sub-images (called as interrogation areas, IA). The correlation was operated between the interrogation area centered on a point aij in the first image and the interrogation area centered on a point bij in the second image recorded with a time δt as [30]: C (m, n) =
∑ ∑ I A1 (i, j) I A2 (i − m, j − n) i
j
where IA1 and IA2 are, respectively, the interrogation areas in the first and in second image of a single exposed image pairs. The maximum peak location in C(m,n) indicates the particle displacement. The velocity vectors are derived from these displacements by dividing them by δt. This procedure was iteratively applied for the whole image by the help of the PivLab algorithm in Matlab. The PivLab algorithm considers pairs of images temporally consecutive by recognizing the lighting intensity reflected by each particle into the flow over time. The cross-correlation is estimated by the fast Fourier transform (FFT) in multi-pass correlation mode [35]. In particular four passes have been used. This means that the estimation of particle’s displacement depends both on the number of frames considered and on the resolution of the recorded images. Finally, the velocity vectors were estimated in nodes of a regular rectangular mesh, identified in the ROI of each image, whose size depends on the spatial resolution (dimension) of the interrogation area (IA). 2.3. Surface Velocity Estimation by PIV Method For each pair of frames, and thus for each corresponding instant tf , the flow velocity components [vx (tf ) and vy (tf )] with respect the orthogonal local reference system (x, y), were determined with the PivLab Tool, as described in the previous Section 2.2. The application of the digital procedure allowed us to estimate, for each image, the distribution of the surface velocity vectors, the stream lines and the vorticity patterns at each time tf . As an example, Figure 3 reports the streamlines superimposed over the images and the velocity vectors superimposed over the vorticity contour maps at time tf = 5 s for images 10, 13, and 20.
[vx(tf) and vy(tf)] with respect the orthogonal local reference system (x, y), were determined with the PivLab Tool, as described in the previous Section 2.2. The application of the digital procedure allowed us to estimate, for each image, the distribution of the surface velocity vectors, the stream lines and the vorticity patterns at each time tf. As an example, Figure 3 reports the streamlines superimposed over the images and the velocity vectors superimposed over the vorticity contour maps at time7tfof=17 5 Geosciences 2018, 8, 383 s for images 10, 13, and 20.
(a)
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(b)
(c) Streamlines superimposed superimposed over over the the images images and and velocity velocity vectors vectors superimposed superimposed over over the Figure 3. Streamlines (1/s)patterns: patterns:(a) (a)Image Image10; 10;(b) (b)Image Image13; 13;(c) (c)Image Image20. 20. vorticity (1/s)
For simplicity simplicity of of exposition, exposition, these these images images have have been been selected selected to to explain explain some some of of the the obtained obtained For results because because they they are are characterized characterized by by equal equal number number of of recorded recorded frames frames and and they they are are located located in in results peculiar positions along the channel. In fact (see Figure 1b), image 10 is located along the straight reach peculiar positions along the channel. In fact (see Figure 1b), image 10 is located along the straight downstream of the inflow images and 20 located of inflow I2ofand inflow reach downstream of the I1, inflow I1, 13 images 13are and 20 areslightly locateddownstream slightly downstream inflow I2 I3, respectively; both the images 20 refer to curved stretches and the image includes and inflow I3, respectively; both 13 theand images 13 and 20 referchannel to curved channel stretches and20the image also the reach characterized by variableby width. From Figure 3 it Figure can be 3observed the flow 20 includes also the reach characterized variable width. From it can bethat observed thatfield the estimated in sectionsin 13sections and 20 reveal confluence of the inflows I2 and I3 and flow field estimated 13 andthe 20 effect revealofthe effect of confluence of the inflows I2 free andsurface I3 and perturbations generated by turbulence forming downstream the confluences It should be free surface perturbations generated eddies by turbulence eddies forming downstreamoccur. the confluences noted that the velocity values reported in Figure 3 have been estimated in 1160 nodes/frame. occur. It should be noted that the velocity values reported in Figure 3 have been estimated in 1160 In the present work, interest is especially devoted to the time-averaged velocity. Thus, by using the nodes/frame. time-series of vx (tf ) work, and vyinterest (tf ), the corresponding time-averaged components (vvelocity. determined x , vy ) were In the present is especially devoted to the time-averaged Thus, by using for each node of the regular mesh identified as a function of the selected spatial resolution of the q the time-series of vx(tf) and vy(tf), the corresponding time-averaged components ( vx ,vy ) were interrogation area. Then, the resultant time-averaged velocity of intensity v = v2x + v2y was estimated. determined for each node of the regular mesh identified as a function of the selected spatial resolution 3. the Results and Discussion of interrogation area. Then, the resultant time-averaged velocity of intensity
2
2
v = v x + v y was
estimated. 3.1. Estimation Error From and the “Introduction” and the results reported in previous works [34,35], it is clear that both 3. Results Discussion the number of frames (which depends on the acquisition time selected) and the dimension of the interrogation 3.1. Estimationarea Error(on which the extension of the data sample depends) could influence the accuracy of the results. The measurement points are nodes of a regular mesh whose dimension depends on the From the “Introduction” and the results reported in previous works [34,35], it is clear that both dimension of the interrogation area and a high dimension of data sample could determine modeling the number of frames (which depends on the acquisition time selected) and the dimension of the processes too much time-consuming. interrogation area (on which the extension of the data sample depends) could influence the accuracy of the results. The measurement points are nodes of a regular mesh whose dimension depends on the dimension of the interrogation area and a high dimension of data sample could determine modeling processes too much time-consuming. In order to perform the sensitivity analysis of the estimated surface velocity values to the number of pairs of frames, for all the recorded images, the velocity components vx(tf) and vy(tf) have been
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In order to perform the sensitivity analysis of the estimated surface velocity values to the number of pairs of frames, for all the recorded images, the velocity components vx (tf ) and vy (tf ) have been calculated in each node of the rectangular mesh, corresponding to a fixed dimension of the interrogation area, by considering six values of the total number of pairs of frames (i.e., of time instants) to be processed (n1 = 100, n2 = 500, n3 = 1000, n4 = 1500, n5 = 2000, n6 = 2400). The standard deviation (estimation error) of the velocity components has been determined for each i-th node of the mesh as: v u 2 8 of 17 u ∑ pi,j − pn j ,i u j=1,n j σn j i = t i = 1, . . . , N n j = 100, 500, 1000, 1500, 2000, 2400 (1) measured averaged value ofnthe examined velocity component p falls in the range pn j ,i ± σ n j ,i and j Geosciences 2018, 8, x u FOR PEER REVIEW
the uncertainty in averaging the component p can be estimated as σE = σ n j ,i / n j [42]. where pi,j and pn j ,i are, respectively, the examined component p [p = v x t f , vy t f ] and the Figure 4 reports, for each component thenumber highestof value of the estimation error obtained by corresponding time-averaged value, N is p,the nodes. From the error theory [42], the σ canexamined be observed that,component especially pfor theincomponent , nthe Equation for eachvalue measured(1) averaged velocity falls the range ppn j ,i= ±v xσntjf,i and n j .ofItthe j ,i √ uncertainty in averaging the component p can be estimated as σ = σ / n [42]. n j ,i jthe lower values. Thus, tends to decrease as the number n j increases; for n j > 1200E it assumes Figure 4 reports, for each component p, the highest value of the estimation error obtained by the maximum value of the uncertainty σE in averaging the component p is obtained for n j = 100. In Equation (1) for each n j . It can be observed that, especially for the component p = v x t f , σn j ,i tends to this case, itashas estimated that thefor uncertainty is less than in values. the x direction andmaximum less than decrease thebeen number n j increases; n j > 1200 σitE assumes the2.4% lower Thus, the > 1200 the uncertainty σ E becomes less than 1% in both the directions 4% in the y direction. For n value of the uncertainty σE jin averaging the component p is obtained for n j = 100. In this case, it
( )
estimated thatathe uncertainty σEisisnecessary less than to 2.4% in the direction and less 4% in xhas andbeen y. This means that large sample size obtain thexsame uncertainty in than averaging the y direction. For n > 1200 the uncertainty σ becomes less than 1% in both the directions x and E j the component p. y. This that of a large sample size is velocity necessary to obtain the same uncertainty averaging the 500, 1000, Themeans deviation the time-averaged components, estimated for n j =in100, component p. 1500, 2000, from those evaluated for the highest number values of the total number of pairs of frames The deviation of the time-averaged velocity components, estimated for n j = 100, 500, 1000, 1500, ( n j = 2400) has been obtained as: 2000, from those evaluated for the highest number values of the total number of pairs of frames (n j = 2400) has been obtained as: v 2 u p − p 2 u n , i 2400, i p − p u ∑ jn j ,i (2) 2400,i t ii==11,N ,N (2) σnσj (npj )p== NN
()
Figure 4. Highest value of estimation error σ (m/s). Figure 4. Highest value of estimation error σ nn j (m/s).
j Figure 5 reports the estimated values of σn j ( p) against n j . From this figure it can be observed that σn j ( pFigure decreases as nvalues for the examined velocity range, low and almost ) exponentially j increases. 5 reports the estimated of σ Thus, n j p against n j . From this figure it can be observed constant values of σn j ( p) are obtained for n j ≥ 1200, i.e., for a number of processed frames equal to or σ n jthan p exponentially that decreases as nofj pair increases. Thus, for the examined velocity range, low greater half of the available number of frames.
()
()
()
and almost constant values of σ n j p are obtained for n j ≥ 1200, i.e., for a number of processed frames equal to or greater than half of the available number of pair of frames.
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(a) (a)
(b)
() p = v y (m/s). Figure5.5.Deviation Deviationσσ ((pp)) for for all all the theexamined examinedimages: images:(a)(a)pp==v v (m/s); (b) pp ==v v (m/s). Figure x (m/s);(b) y (m/s). This behavior can be also observed from Figure 6 where the values of the time-averaged velocity Figure 5. Deviation
σ n j p for all the examined (b) images: (a) p = vx (m/s); (b) nnj j
x
y
This behavior alsopobserved from Figure values the time-averaged velocity v x be and = v y ) estimated for6 where =the 100, 500, of 1000, 1500, 2000, have been components ( p = can n j the This behavior can be also observed from Figure 6 where values of the time-averaged velocity components (p = v x and p = vy ) estimated for n j = 100, 500, 1000, 1500, 2000, have been compared with = 2400. Itfor cannbej seen that for1000, all the considered images, and compared with( pthose evaluated for n j estimated and Itp can = vbe = 100, 500, 2000, have components y )seen those evaluated for= n vj x= 2400. that for all the considered images,1500, and for both the been velocity for both thewith velocity components, well fit the bisector when components, the those points well fit the bisector line when n j seen > 1000. It can be that line for all the considered images, and compared evaluated forthe n j > 1000. n points j = 2400. for both the velocity components, the points well fit the bisector line when n j > 1000.
(a)
(a)
Figure 6. Cont.
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(b)
(
Figure 6. Comparison between the component p = v x ;v y
(b)
)
estimated for
n j = 100, 500, 1000, 1500,
and 2000 and the corresponding value estimated for n j =2400: (a) p = v x (m/s); (b) p = v y (m/s). Figure vy estimated forfor n j =n 100,=500, 1500,1500, and Figure6. 6. Comparison Comparison between between the the component component p p== vvxx; ;v 100,1000, 500, 1000, y estimated j 2000 and the corresponding value estimated for n j = 2400: (a) p = v x (m/s); (b) p = vy (m/s). = v xinterrogation Then, theand sensitivity of surface velocity tofor thendimension has been (m/s); (b) p =area and 2000 the corresponding value estimated v y (m/s). j = 2400: (a) ofp the
(
)
Then, the of surface velocity to the dimension the interrogation area has been been estimatedofby considering different values of performed. Thesensitivity values of the component p have performed. The values of the component p have been estimated by considering different values of Then, the sensitivity to the dimension of thevalue interrogation areaAccording has been In surface this case,velocity it has been assumed a constant the interrogation area (IAk).of n = 2400. the interrogation area (IAk ). In this case, it has been assumed a constant value njj = 2400. According estimated by considering different valuesthe of performed. The[30], values of the to component p have been to the literature literature in order order keep the the background background noise in the the correlation correlation matrix low [42,43], [42,43], to the [30], in to keep noise in matrix low the k). In this case, ait way has been aminimum constant value = 2400. According the interrogation nthej interrogation interrogation areaarea was(IA reduced in such such as to toassumed obtain aa minimum size of of the interrogation area interrogation area was reduced in a way as obtain size area equal to one quarter of the initial size. This is in accordance with literature [42–44] which shows that to theto literature [30],ofinthe order to keep noisewith in the correlation matrix lowshows [42,43],that the equal one quarter initial size. the Thisbackground is in accordance literature [42–44] which the size of the interrogation area should be not less than 4 times of the maximum displacement. Thus, interrogation area was reduced in suchbe a way as to obtain a minimum size of the interrogationThus, area the size of the interrogation area should not less than 4 times of the maximum displacement. the sizetoof of thequarter interrogation areaofof the first in the processing PivLab code defined taking equal one of the area initial size. This ispass ininaccordance withPivLab literature [42–44] which shows that the size the interrogation the first pass the processing code waswas defined taking into into account that the of value of the maximum displacement lmax * dt (where Umax isThus, the the size of the interrogation should be not less than 4 times of the maximum displacement. account that the value the area maximum displacement lmax = Umax *=dtUmax (where Umax is the maximum maximum measurable velocity and dt is the time step). The minimum size of the interrogation area the size of the interrogation areatime of the firstThe pass in the processing code was taking measurable velocity and dt is the step). minimum size of thePivLab interrogation areadefined was assumed was equal to 8value ×is8greater pixels is4greater 4 * lmaxThe = 5.6 size of into assumed account the of thethat maximum lmax =pixels. UmaxThe *size dtmaximum (where Umax is the the equal to 8 × 8 that pixels that than * lmaxdisplacement =than 5.6 pixels. maximum of the interrogation interrogation area was basis of the ROI dimension insize the PivLab code. maximum measurable velocity dtthe isROI the time step). minimum of the interrogation area area was determined ondetermined the basisand ofon the dimension inThe the PivLab code. Thus, the sensitivity sensitivity has been conducted five =5)5) values of spatial resolution was Thus, assumed equal to 8 analysis ×analysis 8 pixelshas that is greater thanfor 4for *five lmax ==5.6 pixels. The maximum size of the the been conducted (k (k values of thethe spatial resolution of of the interrogation area: IA 1 = 32 px; IA 2 = 24 px; IA 3 = 16 px; IA 4 = 12 px; IA 5 = 8 px. It is clear that as interrogation areaarea: was IA determined on the basis of the ROI dimension in the PivLab code. the interrogation = 32 px; IA = 24 px; IA = 16 px; IA = 12 px; IA = 8 px. It is clear that 1 2 3 4 5 thethe size of of IA k decreases, the NNk ofofthe increases. should be Thus, the sensitivity analysis has of been conducted for regular five (k =mesh 5) values of the But, spatial resolution as size IA thenumber number ofnodes nodes the regular mesh increases. But, it it should be k decreases, k considered that a high number of nodes determines modeling processes too much time-consuming. of the interrogation area: IA1 = of 32nodes px; IAdetermines 2 = 24 px; IAmodeling 3 = 16 px; IA 4 = 12 px; IAmuch 5 = 8 px. It is clear that as considered that a high number processes too time-consuming. Figure plots, forall allthe the images, the percent area k% regular againstofof thenumber number of nodes . Figure the size of 77IA k decreases, theimages, number ofpercent nodesarea N k of the mesh increases. But, it kN should be Figure plots, for the IAIA % against the of nodes N . kFigure 7 k 7 indicates that the number of nodes increases with a logarithm law as the percent area decreases. considered high number of nodes determines modeling processes too much indicates thatthat theanumber of nodes increases with a logarithm law as the percent areatime-consuming. decreases. Thus, a reduction offor theall interrogation areapercent implies a remarkable increase of the number of nodes. Figure 7 plots, the images, the area IAk% against ofthe thenumber number ofnodes. nodes Nk. Figure aThus, reduction of the interrogation area implies a remarkable increase of of
7 indicates that the number of nodes increases with a logarithm law as the percent area decreases. Thus, a reduction of the interrogation area implies a remarkable increase of the number of nodes.
Figure Figure 7. 7. Percent Percent area area of of the the interrogation interrogation area area (IA (IAkk%) %) and and the the number number of of nodes. nodes.
Figure 7. Percent area of the interrogation area (IAk%) and the number of nodes.
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For For each each IA IAkk, the cross-correlation has been determined with the PivLab tool and the the velocity velocity components havebeen beenestimated estimatedin ineach eachnode nodeof ofthe thecorresponding corresponding rectangular rectangular mesh. components vvxx(tff) and vyy(t(tff))have Then, Then, the the estimation estimation error error of of the thevelocity velocityvalues valueshas hasbeen beendetermined determinedas: as:
(
)
v 2 u p − p i, k2 u u ∑ k pi,ki, k− pi,k tk p = σi,kσ ( pi ,)k = Nk N k
()
=i =1,1,…, ...,N Nkk k k== 1,…, 1, . . . 5, 5
(3) (3)
where N Nk indicates indicates the the number number of of nodes nodes corresponding corresponding to to the the k-th k-th interrogation interrogation area, area, IA IAk.. Figure Figure 88 where k k , ofthe theestimation estimationerror errorobtained obtainedvarying varyingthe theinterrogation interrogationarea area k. reports the the highest highest value, value, σkσ(kp)p, of reports IAIA k.
()
()
Figure 8. Highest value of σk ( p) (m/s) against IAk . Figure 8. Highest value of σ k p (m/s) against IAk.
From Figure 8 it can be seen that σk ( p) tends to decrease as the size of the interrogation area increases that8ititassumes the lower for 16to pxdecrease < IAk < 20 it increases as the size of Fromuntil Figure can be seen that σvalues as px; the then size of the interrogation area k p tends the interrogation area increases. It should be noted that for IA = 16 px the size of the interrogation increases until that it assumes the lower values for 16 px < IAk 3< 20 px; then it increases as the size of area is reduced of area one half compared to thebe original the interrogation increases. It should noted size. that for IA3 = 16 px the size of the interrogation Thus, the values of each time-averaged velocity component, p, estimated for the interrogation area is reduced of one half compared to the original size. areasThus, corresponding to k = 1, 2, 4 and 5 have been compared with those evaluated assuming k = 3. forbythe interrogation the values of each time-averaged velocity component, p , estimated This comparison is reported in Figure 9 which shows that for k > 2 the points tend to arrange around areas corresponding to k = 1, 2, 4 and 5 have been compared with those evaluated by assuming k = 3. the bisector line, for k ≤ 2 they tend to move away from the bisector line. This behavior suggests that, This comparison is reported in Figure 9 which shows that for k > 2 the points tend to arrange around for the examined case, a reduction of the size of interrogation area of one half compared with the the bisector line, for k ≤ 2 they tend to move away from the bisector line. This behavior suggests that, initial size represents a good compromise between the extension of the data sample and the accurate for the examined case, a reduction of the size of interrogation area of one half compared with the estimation of flow velocity. This result could also be consistent with previous works [45] identifying a initial size represents a good compromise between the extension of the data sample and the accurate limit of the maximum recoverable spatial displacement in any sampling directions to half the window estimation of flow velocity. This result could also be consistent with previous works [45] identifying size in that direction. a limit of the maximum recoverable spatial displacement in any sampling directions to half the Then, in order to verify the reliability of the estimated velocity measurements along the channel, window size in that direction. the flux of mass in adjoining regions of consecutive images was also verified. As an example, Figure 10 reports the comparison between the profiles of the specific longitudinal, mx/ρ = htr v x (htr = local water depth), and transversal, my/ρ = htr vy , fluxes of mass estimated along two adjacent sections. Figure 10 shows that the profiles of the specific flux of mass compare well both in longitudinal and in transversal directions.
()
This comparison is reported in Figure 9 which shows that for k > 2 the points tend to arrange around the bisector line, for k ≤ 2 they tend to move away from the bisector line. This behavior suggests that, for the examined case, a reduction of the size of interrogation area of one half compared with the initial size represents a good compromise between the extension of the data sample and the accurate estimation of flow velocity. This result could also be consistent with previous works [45] identifying a limit of8,the Geosciences 2018, 383 maximum recoverable spatial displacement in any sampling directions to half the 12 of 17 window size in that direction.
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(a)
(b)
(
)
Figure 9. Comparison between , vyy obtained forIAIA , IA IA5 with 2 , 4IA 4 , and Figure 9. Comparison betweencomponent component p p== vvxx ,v obtained for 1, 1IA 2, IA , and IA5 with those those obtained for IA3 : (a) p = v x (m/s); (b) p = vy (m/s). obtained for IA3: (a) p = vx (m/s); (b) p = v y (m/s).
Then, in order to verify the reliability of the estimated velocity measurements along the channel, the flux of mass in adjoining regions of consecutive images was also verified. As an example, Figure 10 reports the comparison between the profiles of the specific longitudinal, mx/ρ = htr vx (htr = local water depth), and transversal, my/ρ = htr v y , fluxes of mass estimated along two adjacent sections. Figure 10 shows that the profiles of the specific flux of mass compare well both in longitudinal and
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0.80
Sec. 5
0.80 0.60
Sec. 5a Sec. 5 Sec. 5a
my/ρ my/ρ
mx/ρ mx/ρ
0.40 0.20 0.20 0.00 0
5
0.00 0
5
10 X (m) 15
(a) 10
X (m) 15
20 20
Sec. 5a Sec. 5
2 1.5
0.60 0.40
25 25
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Sec. 5
2
Sec. 5a
1.5 1 1 0.5 0.5 0 0
5
0
5
0
10 X (m) 15
(b) 10
X (m) 15
20
25
20
25
Figure 10. Comparison(a) of specific flux of mass between sections of adjacent (b)images (sections 5 and 5a, distant 2 cm apart): (a) longitudinal direction; (b) transversal direction. Figure 10. sections of of adjacent images (sections 5 and 5a, 10. Comparison Comparison of ofspecific specificflux fluxofofmass massbetween between sections adjacent images (sections 5 and distant 2 cm apart): (a) longitudinal direction; (b) transversal direction. 5a, distant 2 cm apart): (a) longitudinal direction; (b) transversal direction.
3.2. Discharge Estimation 3.2. Discharge 3.2. Discharge Estimation Based onEstimation the results presented in the previous section, the discharge has been estimated by 3 = 16 px andpresented and previous the surface velocity vectors have been determined, as assuming IA n j = 1500 Based in Based on on the the results results presented in the the previous section, section, the the discharge discharge has has been been estimated estimated by by assuming IASection = 16 px and 1500 velocity explained 2.3,and at nodes theand corresponding mesh. have 33 = 16 px 1500 andthe thesurface surfacerectangular velocity vectors vectors have been been determined, determined, as as assuming in IA nn jj ==of explained in Section 2.3, at nodes of the corresponding rectangular mesh. Then, the transversal sections reported in Figure 1c have been considered and the time-averaged explained 2.3, atsections nodes of the corresponding mesh. Then, in theSection transversal reported in Figure 1crectangular have been considered and the time-averaged v t r and v l velocity components along the reported transversal and 1c thehave stream-wise directions ( time-averaged Then, the transversal sections in Figure been considered andv the velocity components along the transversal and the stream-wise directions (vtr and l respectively—see respectively—see Figure 11) have been estimated at the nodes included directions in each transversal section. v t r aims Figure have been estimated at the nodes included in each transversal section. For( the of the velocity11)components along the transversal and the stream-wise and vl present work, only the stream-wise velocity component v has been considered. has been considered. For the aims of the present work, only the stream-wise velocity component v l l respectively—see Figure 11) have been estimated at the nodes included in each transversal section. For the aims of the present work, only the stream-wise velocity component v l has been considered.
Figure 11. 11. Geometrical Geometrical scheme scheme for for the the discharge discharge estimation. estimation. Figure
Figure 11. Geometrical scheme for the discharge estimation.
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Geosciences 2018, 8, 383 According to
17 reference [10], it has been assumed that the shape of the vertical profile 14 of ofthe stream-wise velocity component is the same at each node of the transversal section. Furthermore, fromAccording the literature it is clear [10], that the depth-averaged can be estimated as a function of the the to reference it has been assumedvelocity that the shape of the vertical profile of free-surface velocity [46] through a coefficient equal to 0.85, which is the value generally used with stream-wise velocity component is the same at each node of the transversal section. Furthermore, other the measurement Recently, Termini and velocity Moramarco applyingasthe entropy-based from literature ittechniques. is clear that the depth-averaged can[6] be by estimated a function of the approach, verified that the maximum velocity is linearly related to the mean flow velocity through a free-surface velocity [46] through a coefficient equal to 0.85, which is the value generally used with dimensionless parameter Φ(M) (M is the entropic parameter) which assumed a value of around 0.8 other measurement techniques. Recently, Termini and Moramarco [6] by applying the entropy-based along the verified curved that stretches of the channel. authors (seetointhe [47]) found value of such aa approach, the maximum velocityOther is linearly related mean flow avelocity through coefficient equal to 0.9. dimensionless parameter Φ(M) (M is the entropic parameter) which assumed a value of around 0.8 present work,ofby into account that(see thein water depths are very low (order of alongInthethe curved stretches thetaking channel. Other authors [47]) found a value of such a coefficient magnitude equal to 0.9.of 1 cm) the difference between the depth-averaged and the surface velocities has been assumed so by that the coefficient has been equal 1. The discharge has been In thealmost presentnull work, taking into account that the assumed water depths areto very low (order of magnitude determined as the sumbetween of the elementary stream-wise fluxsurface per unit time estimated ∂q = vˆalmost l ht r ∂tr of 1 cm) the difference the depth-averaged and the velocities has beenas: assumed
null that the has been assumed equal todirection 1. The discharge has been as ( ∂t r so = width of coefficient the elementary area in the transversal tr—see Figure 11). Indetermined Figure 12 the ˆ the sum of the elementary stream-wise flux per unit time estimated as: ∂q = v h ∂t (∂t = width tr) in r sections r l m 4, 5, estimated values of the discharge (qe) have been compared with those measured (q of the elementary area in the transversal direction t —see Figure 11). In Figure 12 the estimated r 11, 12. From the Section 2.1 it is clear that the value of flow discharge measured by the ultrasonic values of the beento compared with those (qm ) at in sections sections11 4, and 5, 11, instrument at discharge sections 4 (qe) and have 5 is equal QI1 = 0.004 m3/s and measured that measured 1212. is From the Section 2.1 it is clear that the value of flow discharge measured by the ultrasonic instrument 3 equal to QI1 + QI2 = 0.00537 m /s. Furthermore, Table 1 reports the values of the normalized error in at 4 and 5 is equal to QI1 = 0.004 m3 /s and that measured at sections 11 and 12 is equal thesections discharge estimation: to QI1 + QI2 = 0.00537 m3 /s. Furthermore, Table 1 reports the values of the normalized error in the q − qe discharge estimation: σ q = qmm− qe (4) qm σq = (4) qm Figure 11 11 shows shows that that the the points points arrange arrange around around the the line line of of perfect perfect agreement agreement demonstrating demonstrating aa Figure good agreement agreement between betweenthe theestimated estimatedand andthe themeasured measured values discharge. From Table 1 it good values of of thethe discharge. From Table 1 it can can be seen that the highest value of error is of 22.0 (%). be seen that the highest value of error is of 22.0 (%).
Figure 12. Comparison between the estimated and measured values of discharge (l/s). Figure 12. Comparison between the estimated and measured values of discharge (l/s). Table 1. Error in discharge estimation. Table 1. Error in discharge estimation. Section σq qe (m3 /s)
Section 4 5 4 11 5 1211 12
4. Conclusions
σq
0.17 0.17 0.22 0.22 − 0.13 − 0.18 −0.13
−0.18
qe (m3/s) 0.0035 0.0035 0.0033 0.0033 0.0061 0.0063 0.0061 0.0063
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4. Conclusions The present paper concerns the application of fully digital approach for estimating the surface velocity and the discharge. In particular, the efficiency of the procedure is investigated for two parameters: the number of frames to be processed and the size of the interrogation area to be correlated. The analysis has been carried out by the help of the PivLav tool in Matlab. The results obtained from the presented analysis can be summarized as follows: -
-
-
the estimation error of the surface velocity decreases as the number of pairs of frames increases. In particular, for the examined case, the estimation error assumes a low and an almost constant value as the number of processed pairs of frames is greater than 1200 (i.e., equal to or greater than half of the available pairs of frames); the size of the interrogation area plays an important role in the surface velocity estimation. It has been verified that the number of nodes increases with a logarithm law as the interrogation area decreases. But, the problem is that a high extension of the data sample makes the modeling process too time-consuming. As result of the sensitivity analysis of the surface velocity to the size of the interrogation area, it has been found that a reduction of the size of the interrogation area of about one half compared to the initial size represents a good compromise between the extension of the data sample and the accurate estimation of flow velocity; the application of the PIV method has provided detailed information of the spatial distributions of instantaneous surface velocity vectors and of free surface perturbations related to the formation of large eddies downstream of the confluences. Furthermore, by using the spatial distributions of time-averaged surface velocity obtained by applying the PIV analysis, the values of the discharge in peculiar sections along the channel have been estimated. The good agreement between the estimated values of the discharge and the measured ones has demonstrated the ability of the digital image-technique for remote monitoring of free-surface velocity and discharge measurement.
It should be noted that the presented results have been obtained in the controlled environment of a laboratory. This could suggest that the errors in the field might outweigh the errors tested in this study. Despite this limitation, the study allows us to gain insight into the applicability conditions of the PIV method for estimating the free surface velocity and flow discharge in hyper-concentrated flows. The obtained results could also be used for improving the application of the image-technique for monitoring activities of velocity in natural rivers, especially for velocity ranges similar to that analyzed in the present work. Author Contributions: D.T. conceived and designed the work; D.T. and A.D.C. performed the experiments; D.T. and A.D.C. analyzed the data; D.T. and A.D.C. wrote the paper; A.D.C. improved the final editing of the figures; D.T. performed the final correction and editing of the work.” Funding: This research received no external funding Acknowledgments: The experimental apparatus in Palermo has been supported by the Regional Department DRPC and GC of Messina (Sicily). Conflicts of Interest: The authors declare no conflict of interest
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