3.3 Quantitative Image Velocimetry Techniques

Transcription

1 Measurement Technologies in Fluid Mechanics Quantitative Image Velocimetry Techniques Why are these techniques of interest? It is challenging and time consuming to measure flow field properties e.g. u(x, y), ω (x, y) (the vorticity field), u v (x, y) (the Reynolds stress field), ε(x, y) (the turbulent dissipation field). Laboratory measurement of velocity is now routinely done with single point measurement probes: Laser Doppler Velocimeters (LDV), hot-wire or hot-film velocimeters, and Acoustic Doppler Velocimeters (ADV). While it is tedious, in a steady flow it is possible to map out the velocity field. In an unsteady flow, it is extremely difficult, if not impossible! If one is interested in coherent structures or vorticity, the measurement becomes even more challenging with point devices and hence the significant interest in whole field techniques. What are the whole field or quantitative imaging techniques currently in use? Particle Streak Velocimetry (PSV) Laser Speckle Velocimetry (LSV) Particle Image Velocimetry (PIV) Particle Tracking Velocimetry (PTV) Other common names include Pulsed Light Velocimetry (PLV) Particle Image Displacement Velocimetry (PIDV) Particle Displacement Velocimetry (PDV) Digital Particle Image Velocimetry (DPIV) Digital Particle Tracking Velocimetry (DPTV)

2 22 Measurement Technologies in Fluid Mechanics Three dimensional (whole volume) techniques include Holographic Particle Image Velocimetry (HPIV) Holographic Particle Tracking Velocimetry (HPTV) Photogrammetric Particle Tracking Velocimetry (PPTV) What are the Techniques? The common principle to all of the techniques is that the instantaneous fluid velocities can be measured by recording the position of images produced by small tracer particles, suspended in the fluid, at successive instants in time. All of the techniques listed (as well as others) fall into two broad classes, each with different development paths: PIV and PTV. LSV and PIV are different operating modes of the same technique. The velocity field is determined on a regular grid as an average velocity over many tracers contained in a small volume of fluid. These techniques have their roots in solid mechanics. They were originally used to determine the in-plane displacement and strain of solids with diffusively scattering surfaces. PTV and PSV can also be thought of as different operating modes of the same technique. In contrast to PIV, the velocity is determined from the image or images or streaks produced by a single tracer at random locations. These techniques have their roots in the field of flow visualization; particle streak photography and stroboscopic photography. Prandtl (1934) was an early developer of particle tracking techniques, although not the first.

3 Measurement Technologies in Fluid Mechanics PIV the Basics (from Sveen & Cowen, 2004) Let us begin the description of the basics of PIV with a fluid flow that is seeded with particles that can be considered passive tracers, perhaps because they are very small or if they are not so small they are nearly neutrally buoyant with respect to the flowing fluid. A light source, generally a laser light source, is shaped with optical components into a light sheet to illuminate the particles. The light sheet is quasi-two-dimensional in the sense that it is thin in the direction orthogonal to the plane of motion that contains the two components of velocity we are interested in measuring while it is broad in the other two directions. An imaging device, in this case a digital camera, is equipped with appropriate optics (e.g., a lens) to collect images of the particles as they pass through the light sheet. In general, computer controlled timing signals are sent to the digital camera and laser light source (or the optical components that shape the light source into a light sheet) to synchronize the light source to the camera such that discrete images of particles (e.g., short time exposure images) are captured at desired times within each collected image. There are, of course, myriad ways in which particle images can be collected, which fall into two general categories: the single exposure of multiple images and the multiexposure of single images. Again there are myriad ways that the mean displacement

4 24 Measurement Technologies in Fluid Mechanics of particles in any sub-region (henceforth referred to as a subwindow) canbeextracted from the images but the fundamental technique used in multi-exposed single images is autocorrelation analysis while the fundamental technique used in single-exposed multiple images is cross-correlation analysis. As cross-correlation analysis is more straightforward as well as more accurate, we will restrict our discussion of PIV basics to cross-correlation analysis of image pairs that is the exposure of two sequential images, each individually exposed, specifically for analysis by cross-correlation. Let s assume we have collected an image pair where the second image was captured a known time, t, after the collection time of the first image. A typical approach to achieving this is to use a computer to control the details of the camera image integration and the laser firing: The most straightforward approach to cross-correlation analysis is to define a square subwindow with side length N =2 n where n is an integer. N is typically taken as a power of 2 to take advantage of determining the crosscorrelation in the frequency domain via the fast Fourier transform (FFT). As we have discussed N need not be restricted to these discrete values but as this restriction is frequently employed we will assume it for now. Let us assume a value of N = 32 and an image size of NR NC pixels (an acronym for picture element that describes the smallest discrete unit of scattered light intensity measured by a digital camera, sometimes also referred to as a pel ), where NR and

5 Measurement Technologies in Fluid Mechanics 25 NC are the number of rows and columns in the digital image, respectively. The simplest algorithm is to divide each image of the pair into non-overlapping N N subwindows and to then perform the two-dimensional cross-correlation of each subwindow pair in the image pair. The cross-correlation function is defined as R(s, t) = 1 N 2 N 1 i=0 N 1 j=0 F I,J (i, j)g I,J (i + s, j + t) (3.10) where R is the cyclic cross-correlation between subwindows I,J in the first image of the image pair (F ) and the second image of the image pair (G), i, j are the indices of the pixel location within subwindow I,J, ands, t is the 2-D cyclic lag at which the cross-correlation is being computed. As discussed previously, R is often calculated in the spectral domain and hence R(s, t) can be found as: R(s, t) =F 1 [F{F I,J (i, j)}f{g I,J (i + s, j + t)} ] (3.11) where F and F 1 are the Fourier and inverse Fourier transform operators and the star denotes complex conjugate. For the purpose of visualization we consider the example images in Fig. 2a-b (see original paper handed out), which shows the image pair F and G along with the non-overlapping subwindow interrogation grid. Fig. 2c and 2d shows subwindow I =10,J = 15 for each image in the pair along with panel e, which shows the resultant cyclic cross-correlation of this subwindow pair. The correlation plane contains a peak which has a maximum at (s, t) =(26, 25). The displacement is measured from the center of the correlation plane to this peak. The integer displacement in our example is estimated to be dx = 6 anddy = 7. Our simple introduction here raises several issues. First we note that the maximum unambiguous displacement that can be resolved is N/2 pixels. If the displacement is larger than N/2 pixels (but less than N pixels, so at least a few of the particles in the first image subwindow remain in the second image subwindow) the correlation peak will alias to the location ±(N ξ ) whereξ is the actual displacement. If the displacement is larger than N pixels then R represents the cross-correlation of two uncorrelated subwindows and the returned displacement estimate will be the result of a random noise peak (i.e.,

6 26 Measurement Technologies in Fluid Mechanics the lag where a maximum number of particles randomly align themselves between the two subwindows). Secondly, there is an implicit assumption that the particles are being translated without rotation or shear. If the particles undergo rotation and/or shear over the time t between the capture of the two images then we must be concerned about the effect of this rotation and/or shear on the existence of a usable correlation peak. Thirdly, if the flow is not two-dimensional where the out-of- plane motion is identically zero then there is a finite non-zero probability of a particle appearing in the first subwindow moving out of the light sheet (and hence the image) by the time the second image is captured. This out-of- plane motion can clearly lead to a measurement bias if the out-of-plane motion is correlated to the velocity itself. And lastly, if N = 32 our maximum resolvable displacement is just 16 pixels. If we cannot extend this or resolve the displacements to sub-pixel accuracy (e.g., estimate the location of the correlation peak to fractional pixel values) then our maximum accuracy is greater than 3% and our typical accuracy is considerably higher than this. The above example also raises many other questions. With the above primer in PIV as our starting point we will now turn to the details of PIV. 3.5 PIV, The Details Image Sampling The image, I(x, y), is typically sampled with a CCD sensor that integrates the light intensity over a small area, known as a pixel. What do the statistics of a typical PIV image look like? Goodman (1984) showed that for a rough surface, with roughness element length scale > λ (the wavelength of the light source) so phase is a uniformly distributed random variable, that the light amplitude is a circular (complex) random variable. This leads to the fact that the light intensity (intensity equals the square of the amplitude) has a negative exponential probability density function (PDF). Therefore, since we have seen that a seeded fluid flow acts like a rough surface (LSV) we expect PIV

7 Measurement Technologies in Fluid Mechanics 27 to have a negative exponential PDF Log(Number of occurences) Pixel intensity (counts) This will have consequences for PTV, since selecting the threshold value to define particles must be somewhat arbitrary. 3.6 Bandwidth of a PIV Image What is the effective sampling rate needed to yield a good representation of the original image? A signal is band limited if it can be constructed from only signals who have frequency components less than some maximum e.g., its Fourier transform, F (s, t) is

8 28 Measurement Technologies in Fluid Mechanics non-zero over only a finite portion of the frequency domain. Therefore, F (s, t) =0, for s >S, t >T. (3.12) The bandwidth is defined as the maximum of S and T. As we will show, a band limited continuous signal can be reconstructed exactly (given infinite samples) provided that the sampling rate is at least twice the bandwidth. This rate, known as the Nyquist Rate, is max(2s, 2T ). Goodman has shown that the bandwidth of the image intensity for a thin spherical lens image system (aperture-d, focal length-f, coherent light with wave length-λ) is given by: W = D λz 0 = D λf(m +1) (3.13) where z 0 = f(m + 1) is the image distance and M is the magnification, z 0 /Z 0 (Z 0 is the object distance). Now, if we collect an image with illumination wavelength λ = 0.5µm, f # = f/d =8,andM =1wegetW =125mm 1 or the Nyquist rate = 2W = 250 mm 1. For photographic PIV this is not a problem. Most common films (ASA 100) have sufficient resolution and we see that a 1 mm 2 spot must be discretize on a pixel imager. But things don t look good for direct image capture. Most CCD s pixels are about 10µm 10µm, therefore2w =100mm 1 is the best that can be done. However, we do not want to reconstruct the image exactly, we only want to obtain the position of the displacement covariance peak. What we are requiring is not the exact details of the particles shapes (edges) but just their positions (low wave number info!). The covariance function has a spectral density function that is nearly band limited meaning its value vanishes for sufficiently large (s, t) but may not be exactly zero. Therefore F (s, t) 0 for s >S, s >T. Parzen showed that for a 1-D signal the bandwidth, W p with a circularly symmetric spectral density function is defined as the width of a cylinder with the same volume

9 Measurement Technologies in Fluid Mechanics 29 Therefore 1/π W p = s(0, 0). (3.14) Using some math it can be shown that if we sample a square region instead of a circular one and we are interested in the covariance of particle images with diameter d, the covariance width, σ h,canbeshowntobeσ h = d 2/(2.44π) and leads to W p =1/(2πσ h ), so what does this all mean? Given the same optical parameters as above and d = 20µm (σ h =3.7µm). Therefore, 2W p =86m 1, suggesting that pixels over a bit less than 1 mm 2 is sufficient! Note that larger particle image diameters reduce our required sampling rate 30µm images pixels/mm 2. For more details on band limited signals see the paper Westerweel, J. (1993), Analysis of PIV interrogation with low-pixel resolution. SPIE, Optical Diagnostics in Fluid and Thermal Flow, Jul v The Cross-Covariance Estimate The expected value of the cross-covariance ˆR II [s, t] = 1 N 2 N r i N s j F (i, j)f (i + s, j + t) (3.15) where F (i, j) is the first image and F (i, j) is the second image, is shown by Westerweel (1992) to be { } ( E ˆRII [s, t] = 1 s )( 1 t ) R II [s, t] (3.16) N N

10 30 Measurement Technologies in Fluid Mechanics Therefore it is not unbiased (note the bias vanishes as N ). The bias occurs because the shift over (s, t) results in only part of F correlating with F Adrian (1988) reported this as the result of in-plane loss-of-pairs. Inthepresenceofstrong gradients the interpolation region contains more particle pairs with small displacements, therefore, this is an under bias. From the above equation it is apparent that the bias grows linearly with the shift size. Westerweel shows that two conditions arise from looking at the variance of the expected value of ˆR II. He finds that the noise in ˆR II due to random correlations is approximately a uniform random field and goes as 1/N 2. The probability that the noise peak is greater than the correlation peak grows for increasing displacement (s, t). 3.8 Estimate of the Displacement It can be shown that for any non-zero particle image diameter the width of the correlation peak will be greater than 1 pixel. By including values adjacent to the maximum in ˆR II, the center of the peak can be estimated to sub-pixel accuracy. Let s decompose the displacement, (s, t) to s = s 0 + ε s, t = t 0 + ε t (3.17) where s 0,t 0 is the integer displacement and ε s, ε t is the fractional part. Therefore 0.5 < ε s < < ε t < 0.5 (3.18)

11 Measurement Technologies in Fluid Mechanics 31 In the absence of an estimate for ε s, ε t the error is ± /2 where is one pixel. N = 32 a displacement of 8 pixels has an uncertainty of ±0.5/8 6%. For If we digitize the same sub-window into 256 pixels (N = 256) the uncertainty is now ±0.5/64 1%. Therefore to work at small N, we need sub-pixel estimates of the center of the covariance peak. It can be shown that covariance {ˆε s, ˆε t } = 0 (i.e., ˆε sˆε t =0), therefore, we can work with the 1-D problem without loss of generality. For narrow correlation peaks (N d/ and s, t N) the covariance width is small enough that only the nearest pixels to the peak contain significant information Since we generally satisfy the conditions for narrow covariance peaks we will look at 3-point sub-pixel estimators. Center-of-Mass (COM) ˆε C = Ignores need for a peak since ˆR 0 can be less than ˆR 1 or ˆR +1. ˆR +1 ˆR 1 ˆR 1 + ˆR 0 + ˆR +1 (3.19) Parabolic ˆε P = ˆR 1 ˆR +1 2( ˆR 1 2 ˆR 0 + ˆR +1 ) (3.20) Requires a peak as 2 ˆR 0 must be > ˆR 1 + ˆR +1 if we require a local maximum and not a local minimum.

12 32 Measurement Technologies in Fluid Mechanics Gaussian The log of a Gaussian curve is parabolic, therefore ˆε G = ln ˆR 1 ln ˆR +1 2(ln ˆR 1 2 ln ˆR 0 + ln ˆR +1 ) (3.21) Note, again ˆR 0 must be a peak and ˆR i > 0 i = 1, 0, 1. Clearly all 3 are a balance of ˆR 1, ˆR+1 with a normalizing value in the denominator. Note that ˆε C will always yield a result and ˆε G has the most restrictions. Therefore we expect ˆε C to be the most robust and ˆR G the least. On the other hand, ˆε C does not even acknowledge that we have a peak while ˆε G acknowledges and uses shape information. Therefore we expect ˆε G to be the best performer and ˆε C the worst. What is the expectation of these estimators? Let s begin with the bias by ignoring the fractional displacement for a moment. Therefore R 1 = R +1 = a 1 R 0 where 0 <a 1 < 1. Note that a 1 = R ±1 /R 0 and is proportional to the width of the correlation peak. After some math (see Westerweel, 1992) a 1 1+2a 1 for ˆ ε C E{ˆε} = 2 N N N m 0 a 1 4(1 a 1 ) for εˆ P (3.22) 1 4 ln(1/a 1 ) for εˆ G Clearly the {}terms are simply constants. But note that the coefficient to the bracket

13 Measurement Technologies in Fluid Mechanics 33 is a function of the displacement m 0 (the integer pixel displacement). This is a negative bias, the result of smaller velocities leading to a higher probability of pairing. If PIV is to operate in an unbiased manner this must be corrected for. The correction is: where F I = R II[s, t] = ˆR II [s, t] F I [s, t] ( 1 s )( 1 t ) N N (3.23) (3.24) Note: This could have been predicted looking at the equation on In PTV the typical bias is ignored as the tracking of particles will remove the bias. Now let s consider the fractional part. A particle will now have a small imbalance in R 1 R +1 therefore we will model the particle image as: R 1 = a 1+ε 1 R 0 = a ε 1 R +1 = a 1 ε 1 (3.25) Note that this is not Gaussian! Westerweel (the guru of analytical PIV!) shows that ˆε C = ˆε P = ˆε G = sinh(ε ln a 1 ) cosh(ε ln a 1 )+ 1 2 aε 1 1 sinh(ε ln a 1 )/2 cosh(ε ln a 1 ) a ε 1 1 ε 2 2ε (3.26) (3.27) (3.28) (3.29) (3.30) Note that the Gaussian sub-pixel fit is independent of a 1, C.O.M. and parabolic are f(a 1 ).

14 34 Measurement Technologies in Fluid Mechanics If we look at the RMS error 1/2 1/2 (error)2 dε we get: Therefore Gaussian performs the best under all particle diameters. 3.9 Post-Processing: Stray Vector Removal Two approaches, temporal statistics and spatial statistics. Cowen & Monismith 1997 use temporal statistics and the assumption of stationarity (or repeatability) but spatial statistics are the norm and have the added advantage that they work on individual vector fields so let s begin there. The principal reference is: Westerweel, J., 1994, Efficient Detection of Spurious Vectors in Particle Image Velocimetry Data, Exp. Fluids, 16: Detectability It would be nice if we could simply look at a correlation-peak (PIV) and decide based on some criteria whether or not we think it is valid. Keane G. Adrian (1990) proposedthe detectability, D 0, as such a measure. They define the detectability as the ratio of thepeak value of the first correlation peak to the second and suggest that if D 0 is greater than that the vector is valid. Westerweel demonstrates that any of three methods based on the statistics of the measured vectors in a field perform better.

15 Measurement Technologies in Fluid Mechanics The Global Mean If v i,j is the observed vector at i, j then the global mean v1 = 1 N i,j v ij. The variance of this statistic will be σε 2 + σ2 v where σ2 v is the variance of the actual velocity field and σ2 ε is the variance of the error. So, one possibility is to try to select the allowable variance of the signal and remove outliers. However, σε 2 σ2 v in practice so this is hard The Local Mean Same idea now over a small neighborhood v ij = 1 N m ( k,lεm v i+k, j+l v i,j ) Consider 3 3 region (N m =8). Itturnsoutyouneedtoknowaprioriwhich vectors are bad to make reasonable decision criteria, if you can, perhaps using global mean. This performs fairly well Local Median Same idea as local mean but use the median. The advantage is that the median is much more robust to outliers, eliminate the need for aprioriinformation on which vectors are bad. Westerweel finds that this performs the best.

16 36 Measurement Technologies in Fluid Mechanics Adaptive Gaussian Filtering Given temporally stationary and/or spatially homogeneous data we can use the statistics of large data sets to filter the data itself. Consider the histogram of this data initially: we see any noise will result in a larger standard deviation than the actual random process of interest (turbulence) possesses. Our goal is to winnow the data down so that we retain our best estimate of the mean ±ασ where α is a coefficient that captures the range of the data in terms of a multiple of the standard deviation where clearly the range grows as the number of data points in our sample grow. Following Cowen & Monismith (1997), we use an iterative (adaptive) approach where in the first iteration we estimate the mean with the median (as it is more robust) and we estimate the standard deviation using a robust estimator based on what is known as the inter quartile range (discussed later this semester). We then set α based on the the expected multiple of the standard deviation that a single data point in a set of N data points sits from the mean, found based on the probability p = 1/N. The value of p is found from Student s t distribution (which is essentially a Gaussian distribution of N> 30) using say tinv in Matlab. Just to be safe a factor of safety is used in the iteration scheme so that α is initially set at twice the expected value in the first iteration and then {1.25, 1.15, 11.06, 1.03, 1.01} in each subsequent iteration with the factor of safety being 1.00 for all remaining iterations. In remaining iterations we typically switch to the actual mean and actual standard deviation of the data. Note I have coded this up in the.m file agw filter which is available in the sample data link on the course web site but I have used robust estimators all the way through as the data we are getting in class sometimes is a bit more noise dominated than our research data. You can easily modify this code to use the mean and standard

17 Measurement Technologies in Fluid Mechanics 37 deviation if you chose. Note for multicomponent data we typically make the decision that if the data is filtered in one component we throw out all components at that instant in space or time. The AGW filter I have provided has four inputs and two outputs where thetwooutputs are overwritten into two of the inputs (i.e., Data and Time): (Data,Time)=agw filter(data, Time, DataMin, DataMax). DataMin and DataMax are what is known as threshold filters they are the minimum and maximum values that you will allow the data to contain. For ADV data you might set these at the min and max velocity based on the velocity range or if you have some knowledge of the flow you could perhaps set this a little more narrowly but the idea is to use DataMin and DataMax only to remove data that is clearly an outlier, as in it simply cannot be data from the physical process you are studying and it must be due to noise. Your goal is to remove the outliers (if any) that are so large that the estimates of the standard deviation in the AGW filter will work adequately. I always start with these set extremely wide based on velocity range for ADV and the minimum and maximum velocity that can be determined from PIV based on how the subwindows are set up. You can always turn this trheshold fitler off by setting the values to extreme values, for example DataMin= 1000 and DataMax= Then, if your AGW filter is not converging, start to look at histograms and see if you need to set DataMin and DataMax more carefully. Note that my version of the AGW filter tracks the row indices of the input data in the array Data (the code assumes the longer dimension is the time dimension and then transposes the matrix so that this is the row dimension if necessary), which corresponds to a specific time in the input vector Time. Then, as data is flagged spurious by the AGW filter processes the indices of the spurious data are tracked and removed from both the Data and Time array and vector, respectively, so that the final results returned in Data and Time are no longer regularly spaced in time there are gaps in time where spurious data was removed. This is not a problem for calculating most metrics (e.g., mean, variance, etc.) but if you need continuously sampled data (e.g., spectra, autocorrelation),

18 38 Measurement Technologies in Fluid Mechanics you will need to interpolate the missing data back in. I recommend just using a linear interpolation method for now so as not to introduce spurious noise into the data set. We may revisit this interpolation process later in the semester as there has been research on how to optimally carry out this interpolation. Further note that if any individual component in Data is flagged bad at a specific time all of the Data at that Time value or removed as well as that Time value. Thus the Reynolds stress can still be calculated by multiplying the instantaneous values in Data and taking the appropriate averages