Improvements achieved with MP-PIV by resolving splintered peaks have been quantified through Monte-Carlo analyses. Assessments have been performed on synthetic images involving two different velocity fields: a boundary layer of increasing free-stream velocity, tested for different seeding densities, and a sinusoidal field of varying frequency, tested for different amplitudes. Synthetic images were generated following the approach described in Lecordier and Westerweel (2004) using Gaussian shaped tracer images, with diameters drawn from a normal probability \(\mathcal {N}(\mu ,\sigma ^2)=\mathcal {N}(3{\mathrm{\ pixel}},1{\mathrm{\ pixel}}^2)\). The mean intensity of particles was 50% of the maximum gray scale and the standard deviation was 18%. Intensity profiles were integrated over pixel elements adopting a pixel fill-ratio of unity and discretised in 16 bits. The synthetic images were not altered with artificial noise , such that observed differences in performance could be ascribed solely to algorithmic variations..
MP-PIV was tested and compared to four different PIV techniques. The first technique, referenced hereafter as MGRID, is similar to the multi-grid algorithm of Scarano and Riethmuller (2000) but without any vector validation. In a slightly modified version of MGRID, MGRIDF, a SNR filter is introduced to automatically exclude vectors with a SNR below 1.3. MGRIDF is thus identical to standard multi-grid methods. In order to investigate the improvements in terms of spatial resolution, intensity weighting as suggested by Nogueira et al. (2001) has been included and is referred to as LFC (local field correction). Finally, a PIV algorithm adopting cross-correlation between windows of different sizes (DSIW, different sized interrogation windows) was also considered as described by Liang et al. (2002). An overview of the considered algorithms in terms of their difference in settings is given in Table 1.
Table 1 Summary of the algorithms used in the numerical assessment
Boundary layer
The displacement field (u, v) related to a laminar boundary layer on flat plate was simulated by resolving Prandtl’s equations (Prandtl 1938) in the vicinity of a wall, i.e. Blasius’ solution;
$$\begin{aligned}&2\frac{\mathrm{d}^{3}f}{\mathrm{d} \eta ^3}+f\frac{\mathrm{d}^2 f}{\mathrm{d} \eta ^2}=0 \quad \text{ with } \quad f(0)=\frac{\mathrm{d}f}{\mathrm{d} \eta }(0)=0 \quad \text{ and } \quad \underset{\eta \rightarrow \infty }{lim}\frac{\mathrm{d}f}{\mathrm{d} \eta }(\eta )=\infty \nonumber \\&\quad u=U_{\infty }\frac{\mathrm{d}f}{\mathrm{d} \eta }, \quad v=\frac{1}{2}\root \of {\frac{\nu U_{\infty }}{x}}\left\{ \eta \frac{\mathrm{d}f}{\mathrm{d} \eta }-f\right\} , \quad \eta =y\root \of {\frac{U_\infty }{\nu x}} \quad \text{ and } \quad \delta =\frac{5.0}{\root \of {U_{\infty }/\nu x}}. \end{aligned}$$
(2)
The free-stream displacement \(U_\infty\) was adjusted from 0.5 to 33 pixels with increments of 0.5 pixels. A total of 5000 independent simulations were performed for each displacement field. The simulations were repeated for three seeding densities: 0.03, 0.1 and 0.3 particles per pixel (ppp). The boundary layer thickness \(\delta\) was set to a constant 64 pixels and the error analysis was focused on an image portion of 65-by-65 \(\mathrm{pixels}^2\) in size adjacent to the wall. Window sizes WS were reduced from 65 to 17 pixels in three refinement steps. Imposing a 75% window overlap, a final displacement field with a constant vector spacing of 4 pixels was obtained giving a total of 225 vectors. A plot of the velocity profiles is presented in Fig. 7.
Due to the potential appearance of multiple valid correlation peaks, the single resulting velocity vector is highly dependent on the underlying flow and instantaneous seeding distribution. To quantify the correspondence between the imposed ideal \((u_{{\mathrm{id}}},v_{{\mathrm{id}}})\) and measured \((\hat{u},\hat{v})\) displacements, results are shown in terms of normalised root of the averaged mean square error in the velocity components (\(\epsilon\), hereafter referred to as error) (Schluchter 2005). This heuristic does not distinguish between bias and random error but is indicative of how representative the measurement is of the underlying flow, and was evaluated as (\(N=1000\times 225\)):
$$\begin{aligned} \varepsilon = \frac{1}{U_{{\mathrm{ref}}}}\sqrt{\frac{1}{N}\sum _{i=1}^{N}\left[ \frac{\left( u_{{\mathrm{id}}}-\hat{u_i}\right) ^2+\left( v_{{\mathrm{id}}}-\hat{v_i}\right) ^2}{2}\right] }, \end{aligned}$$
(3)
where \(U_{{\mathrm{ref}}}\) is the spatially averaged velocity \(U_{{\mathrm{ref}}}=\frac{1}{N}\sqrt{\sum _i \left( u_{{\mathrm{id}}}^2+v_{{\mathrm{id}}}^2\right) /2}\). Plots of the measurement error are presented against the free-stream velocity \(U_\infty\) and the maximum rate of strain S;
$$\begin{aligned} S_{\max }\equiv \frac{\partial u}{\partial y}\Bigr |_{y=0}=\frac{5.0 U_\infty }{\delta }\frac{\mathrm{d}^2 f}{\mathrm{d} \eta ^2}\Bigr |_{\eta =0}\equiv \frac{1.6605 U_\infty }{\delta}. \end{aligned}$$
(4)
Equation 4 expresses the velocity gradient in terms of pixels/pixel, which, given the imposed values, ranged from 0.013 pixels/pixel (\(U_{\infty }=0.5\, \mathrm{pixels}\)) to 0.85 pixels/pixel (\(U_{\infty }=33\, \mathrm{pixels}\)).
Figure 8a presents the results of the error analysis for a seeding density of 0.03 ppp. For such a small seeding density, DSIW is unable to produce a reliable correlation map and produces noisy results even for the smallest gradients tested. By increasing the free-stream velocity, \(\epsilon\) for DSIW appears to initially decrease. However, the reader should note that the absolute error (\(\epsilon U_\mathrm{ref}\)) is actually increasing. The observable tendency is only due to the adopted normalization in Eq. 3. The other image processing approaches present a constant normalised error \(\epsilon\) of approximately 0.03 up to a gradient of 0.2 pixels/pixel. After this point, MGRID, MGRIDF and LFC present a sharp increase in error while MP-PIV remains constant. When gradients are increased up to 0.5 pixels/pixel, the normalised error for MP-PIV increases to 0.10 compared to 0.20 (MGRIDF), 0.25 (LFC), 0.32 (MGRID) and 0.36 (DSIW). Increasing the velocity gradients even further from 0.5 to 0.6 pixels/pixel, standard PIV algorithms produce error levels which are more than 20% the \(U_{{\mathrm{ref}}}\) whereas the error with MP-PIV remains below 20% up to a gradient of 0.7 pixels/pixel.
In Fig. 8b, the curves for a seeding density of 0.1 ppp show a similar behaviour to the case of lower seeding density (Fig. 8a), with the biggest difference being the change in error with DSIW. For this seeding density, the reliability of DSIW has improved, yielding error levels on par with the other methodologies tested. It is worth noting the difference between MGRID and MGRIDF. The mere implementation of a SNR threshold in MGRIDF allows a reduction of the error when dealing with less severe gradients, confirming the detrimental effect of multiple correlation peaks if left untreated. However, at higher gradients imposing a SNR threshold inhibits the detection of a valid correlation peak among multiple peaks of (near-)equal magnitude, increasing the overall error. Best results are obtained by MP-PIV with lowest error levels throughout the entire range of gradients tested. LFC, which aims to enhance spatial resolution, produces error levels which are slightly lower than the other techniques tested, but only in the lower range of gradients (between 0 and 0.25 pixels/pixel). The adoption of a weighting window is well-documented to increase the spatial resolution of the measurement (Astarita 2007; Nogueira et al. 1999). Current results show though that weighting can be harmful if the linear size of the window is not increased accordingly. Particles belonging to the periphery of the window will contribute less to the correlation peak, producing worse results in case of high amplitude displacements. This observation is confirmed by the error levels produced by LFC in case of high gradients. Figure 8a–c show that the error generated by LFC for gradients higher than 0.5 pixels/pixel can be more than twice the error produced by the other techniques.
With increasing seeding density error levels generally reduce. At a seeding density of 0.3 ppp (Fig. 8c) DSIW produces results which are slightly better than MGRID, MGRIDF and LFC, especially in case gradients are higher than 0.5 pixels/pixel. However, also in this case, MP-PIV obtains the lowest error levels across almost the entire range of gradients tested, confirming it being the most suitable methodology for complex velocity fields.
Sinusoid test
To characterise the behaviour and improvements of MP-PIV even further, the algorithms’ response to a one-dimensional sinusoidal displacement field was considered:
$$\begin{aligned} u_{{\mathrm{id}}}(x,y) = A\sin \left( y \frac{2\pi }{\lambda } + \phi \right) . \end{aligned}$$
(5)
Sinusoidal displacements were imposed by adjusting the normalised wavelength from 0.25 to 1.5 and randomly selecting the phase \(\phi\) in a Monte-Carlo fashion. The normalised wavelength was defined as the ratio between the smallest correlation window size \({\rm WS_{\min }=17 pixels}\) and imposed sinusoidal wavelength \(\lambda\); \(\lambda ^* ={\mathrm{WS}}_{\min }/\lambda\). Amplitudes A of 3, 5 and 7 pixels were imposed. Measured displacement fields were fitted with a sinusoid and the reconstructed amplitude \(\hat{A}\) was normalised with the imposed A producing an evolution in normalised amplitude \(\hat{A}/A\) with normalised wavelength \(\lambda ^*\). For all sinusoids tested, the seeding density was kept at a constant value of 0.1 ppp. Images were analysed once again with initial window sizes of 65 pixels and iteratively halved to 17 pixels. A mutual window overlap of 75% was set to obtain a final vector spacing of 4 pixels.
For a sinusoid of 3 pixels of amplitude, results are presented in Fig. 9a. The plots show that only for small amplitudes the frequency response of cross-correlation resembles the expected behaviour of a moving average (Scarano 2003; Theunissen 2012). MP-PIV performs slightly better compared to the other approaches, except for LFC. For this specific velocity field, the spatial resolution obtained using LFC is superior to the other techniques tested, with values of normalized amplitude higher than 0.5 up to \(\lambda ^*\approx 1.1\). This increased spatial resolution is to be expected as the advantages of using a weighting functions have already been shown in the literature (Astarita 2007). From a practical perspective however, smaller amplitudes demand reduced temporal separation between snapshots, which as advocated is not always optimal. When the sinusoid amplitude is increased from 3 to 5 pixels (Fig. 9b), LFC begins to show the same deficiencies as presented in Sect. 3.1 with an increased amplitude modulation for \(\lambda ^*<0.6\) compared to the other analysis approaches. At \(A=5 \mathrm{pixels}\), MP-PIV behaves slightly better than the other methods tested, offering a marginally higher spatial resolution in the range \(\lambda ^*=0.3\)–0.7. Increasing the sinusoid amplitude even further (Fig. 9c) to \(A=7 \mathrm{pixels}\), the frequency response of all the algorithms becomes completely different from the standard moving average assumption, confirming the strong non-linearity of the PIV algorithm (Theunissen 2012). When a sinusoid of 7 pixels amplitude is analysed, MP-PIV presents the best spatial resolution among all the algorithms tested, with LFC showing the worst spatial resolution for the same reason as described in Sect. 3.1.
Computational effort
An absolute measurement of the computational effort for MP-PIV is impossible to provide since it is entirely dependent on the complexity of the flow. In case of simple velocity fields (i.e. flow gradients are such that correlation maps contain a single peak), the computational overload is only constituted by the vector positioning process. All the parameters being equal to a standard PIV algorithm (i.e. correlation window size, vector spacing, number of iterations, interpolation technique, etc.), the additional overload for MP-PIV slows down the analysis by a factor of 1.5. However, such a comparison is unfair. In fact, MP-PIV produces a denser displacement field from the first iteration and could potentially reduce the number of iterations required for the predictor-corrector to converge, saving computational time compared to a standard PIV algorithm.
When multiple peaks are present, the computational overload depends on several factors, including but not limited to: the quality of the images, the seeding density, the complexity of the flow, the sensitivity (\(\sigma _\mathrm{thr}\)) chosen by the user, etc. For very complex interrogation windows (more than five peaks detected), a computational time up to six times higher was experienced. However, the reader should note that the additional computational time for a single correlation window does not necessarily imply a slower analysis for the entire image, since a slower but more accurate displacement field in the first iteration allows a potential reduction in the required number of successive iterations.
Figure 10 presents the computational time for MP-PIV and other techniques described in Table 1 for a pair of synthetic images of a boundary layer as described in Sect. 3.1. As the amplitude of the boundary layer increases, the flow becomes more complex and the computational time of MP-PIV rises as advocated. MP-PIV has been tested for three different seeding densities, 0.03, 0.1 and 0.3 ppp (particles per pixel) to show the dependency of the computational time on the amount of particle images. As the seeding becomes denser, the underlying flow is better sampled and correlation maps will contain more information, enabling an increased number of correlation windows to be elaborated by MP-PIV and rendering a more accurate flow field (cf. Fig. 8).