10.3 DDPIV version 4.6.0c R14 IPP
10.3.3 FINDFLOW
of refraction of the medium relative to air. This is a simplified version of the correction depicted in equation 20 ofPereira and Gharib[2002], since by doing the dewarping in the fluid allX andY dependency of the correction is removed.
FINDPART is the key step in the processing because its result is the reconstructed point cloud corresponding to the locations in space of the particles that were imaged. Typically vector fields are obtained using particle tracking algorithms and thus the error introduced in this step propagates directly to those results. Great care has been taken to ensure that the calibration is accurate and that the hardware design is such that once a calibration is obtained the results are consistent from power cycle to power cycle. Chapters 12, 13, and14 examine in detail the simulated and experimental precision of the key parts of the reconstruction performed inFINDPART.
10.3.3.1 PIV
The cross-correlation scheme first divides the domain, or flow cell, into a three-dimensional grid of volume elements called voxels. Typically they are laid out such that the overlap between voxels is 50% of the dimension on each side.
The correlation itself is computed by creating a three dimensional Gaussian for each pair of particles whoseX, Y, Zposition is the midpoint between the two particles’ position—that is, if there areM particles in the voxel for the first frame andN particles for the second, thenM×N Gaussians are formed. During FINDPART, each particle is assigned a radius equal to the average of the 1/e2 radii of the particle images that collectively matched to form that particle. DuringFILTERPARTthis radius can be converted to real units using a sizing calibration and application of the Mie-scattering- based sizing algorithm. The pair Gaussians formed for the PIV calculation have standard deviation equal toq
r2i +R2j, whereri is the radius of the particle in the first frame andrj that of the one in the second. The intensity of the pair Gaussian is set ase−a2, where a= (rj−ri)/ri×100.
Once all the pair Guassians are defined, they are added together yielding a three-dimensional correlation space, and thelmqnfunction from thePDLnonlinear optimization library is used to find theX, Y, Zcoordinate of the maximum of this sum. The resulting velocity for the voxel has a vector pointing from the center of the voxel to the location of the maximum of the correlation space and a confidence defined as the maximum value of the correlation divided by the sum of all the pair Gaussian peak values divided by the square root of the number of pair Gaussians.
The PIV method of calculating velocity has been more or less abandoned because it is extremely sensitive to ghost particles and requires several particles per voxel to yield a good vector, which typically implies only 1,000 or so vectors per pair. Moreover rarely are the vector fields presentable without outlier correction (see below) and smoothing.
10.3.3.1.1 Outlier Correction An erroneous vector in a calculated vector field is known as an outlier. They can be caused by several factors, such as the voxel size being too small relative to the displacement, the number of particles in the voxel being too low, or too much noise being generated by ghost particles.
One of the simplest methods for finding such vectors in 2D PIV is to compare each vector the average of its eight immediate neighbors. If the vector deviates more than a certain threshold from this mean, it is deemed an outlier, and can be removed (the preferred route) or replaced by the average.
With three dimensions there is the added advantage that now each vector has 26 immediate neighbors. In DDPIV, outliers are replaced by the weighted average of their 26 neighbors. If the vector checked is assignedi, j, kcoordinates of 0,0,0, then neighbors that have|i|= 1 andj, k= 0,
|j| = 1 andi, k = 0, or |k| = 1 andi, j = 0 (that is, the 6 next-door neighbors, or centers of the
faces of the 3 by 3 cube) are given a weight of 1. Those with |i|,|j|= 1 andk= 0,|j|,|k|= 1 and i= 0, or |i|,|k|= 1 andj= 0 (the 12 in-plane diagonal neighbors, or the centers of the edges of the 3 by 3 cube) are given a weight of√
2/2, and those with|i|,|j|,|k|= 1 (the 8 remaining corners of the 3 by 3 cube) are given a weight of√
3/3.
This correction can be applied in a multiple-pass fashion, however, it is dangerous and can make any random noise vector field resemble turbulent free-stream flow.
If outlier correction is turned on there is the further option of calculating a refined flow estimate, also known in 2D PIV as window shifting. In this scheme, the first cross-correlation is used as an initial guess. For the second pass, the voxel is shifted in the second frame by the amount indicated by the first vector and the cross-correlation is performed again. The final velocity is the vector sum of these two vectors. The idea behind it is that in all flows it is likely that particles are leaving the voxel between frames, so by shifting the voxel the second cross-correlation should be much stronger than the first.
10.3.3.2 PTV
A recent addition to the software were three particle tracking algorithms originally written by Hein- rich St¨uer and implemented directly by Francisco Pereira as detailed inPereira et al.[2006c].
In particle tracking, the idea is to finish with each particle in the first frame connected to one in the second frame. The three algorithms differ in the way they decide which particle pair is a correct match. However they all have in common that the match of a given particle in the second frame must be within some search radiusRscentered about an initial guess which can be 0, the center of mass of the particles in the voxels, the vector obtained through a coarse-grid PIV calculation. Rsis designated as a percentage of the voxel size, thus allowing the user to bias the search by making the longest dimension of the voxel correspond to the direction of the mean flow in the flow cell. This can be very helpful in cases of very directional flow, such as a jet.
10.3.3.2.1 Nearest Neighbor This is the simplest of all the algorithms, and simply chooses the destination particle as the one which is closest in space to the source particle. Its implementation is not quite that simple because it does verify that the chosen vector is the best possible vector for both particles.
10.3.3.2.2 Neural Network The neural network implementation is based on that of Labont´e [1999] with the modification that it is used as a first step in a modified nearest-neighbor scheme.
It is computationally intensive and its results did not show exuberant promise in initial testing and thus it is not currently used. It is compared to the other two schemes inPereira et al.[2006c].
10.3.3.2.3 Relaxation method 3
The relaxation method is the method of choice, it being nearly as fast as nearest neighbor in most cases but much less susceptible to ghost particles. The key of this method is that neighboring particles are assumed to have similar displacements between frames.
The user designates a neighborhood, Rn, which will enclose all the particles in the first frame whose vectors will be compared according to a semi-rigidity condition. As with the other algorithms, Rsis the search area for destination particles in the second frame, centered about the source particle plus the initial guess vector. The relaxation method also adds the parameter Rq, which is the magnitude of the maximum allowable difference between vectors emanating from particles that are within Rn of each other. As with Rs, Rn and Rq are defined as percentages of the voxel size.
Figure 10.3-2depicts this arrangement in space. The current source particlei, shown with a thick red outline, marks the center of the neighborhood Rn. TheNn particles in the first frame within Rn (black circles) will provide a reference point for the semi-rigidity condition. The initial guess (thick grey vector) shifts the center of the regionRs. The Nsparticles in the second frame that fall withinRs (cyan circles) can be possible links for the current source particle. Note that there are particles like particlel in the second framenot withinRs(light cyan circles) which may be linked to particles in the first frame that are withinRn—these are simply links from a previous iteration on those particles.
Figure 10.3-2: A graphical representation of the relaxation method. See the text for a detailed description.
voxel size
Rn
Rs
Rq
initial guess
neighbor link current link
i
j l
i,j k,l
k
In the figure, the link labeled “neighbor link” between particles k andl is shown again in grey translated so that its tail coincides with the tail of the current link between particlesiandj. Thus the green vector shows the difference between links k, l and i, j. Centered about the tail of this
3The discussion presented here follows closely that ofPereira et al.[2006c] and uses similar notation.
difference is the volume Rq, against which the difference vector is checked. In this case the green vector clearly exceeds the bounds ofRq thus the pair of links being checked is “bad”. For each pair of links checked, weights are assigned such that
Qi,j,k,l=
1 ifkdi,j−dk,lk< Rq 0 otherwise
(10.3-4)
where di,j is the vector of the link i, j dk,l is that for the link j, k. In the implementation it is a component check—that is, if any component of the vector exceeds that dimension of Rq it is considered “bad”. An additional weight can be assigned by comparing the link i, j to the flow estimate:
Fi,j=
1 ifkdi,j−uc,i∆tk< Rq
0 otherwise
(10.3-5)
where uc,i is the velocity estimate for particle i. The actual value of uc,i is computed by taking a Gaussian-weighted average of the estimated velocity field (provided by center-of-mass or cross- correlation on a voxelized grid, as mentioned above) where the center of the Gaussian is at particle iand the standard deviation is the largest diagonal ofRs.
These weights are used to favor links that are “good” in a probability assigned to a link which indicates the likeliness that it is the best link for a given particlei within a neighborhoodRn.
For iteration a, the probability is updated according to equation 10.3-6, which is a modified version of the probability presented inBarnard and Thompson[1980]:
P˜i,j(a)=Pi,j(a−1)
A+B
Nn
X
k=1 Npk
X
l=1
Pk,l(a−1)Qi,j,k,l
+CFi,j
(10.3-6)
whereNpk is the number of possible links of a neighbor particle kto the second frame (the number ofk, llinks). The constants are fixed asA= 0.3 andB= 3.0 inBarnard and Thompson[1980] and the constant C is defined inPereira et al.[2006c] to beC = 1.0. In words, if the linki, j does not satisfy the conditions set in equations10.3-4and10.3-5then it is severely punished. The strongest reward comes from the probabilities of particle i’s neighbors’ links which satisfy the semi-rigidity condition of equation10.3-4.
The distinction between ˜P and P is that ˜P is not normalized, that is, ˜P does not satisfy
Pi?+
Npi
X
j=1
Pi,j= 1 (10.3-7)
for a given iteration. That is, the sum of the probabilities of each link emanating from particle i plus the probability that there is no link from particleimust equal 1. ˜P is normalized at iteration
aby
Pi,j(a)=
P˜i,j(a) Pi?(a−1)+PNn
j=1P˜i,j(a)
(10.3-8) andP? is set by
Pi?(a)= Pi?(a−1) Pi?(a−1)+PNn
j=1P˜i,j(a)
(10.3-9) Initially the probabilities are set to
Pi,j(0) =Pi?(0)= 1
Npi+ 1 (10.3-10)