In the present study, a simple method is developed to apply astigmatism particle tracking velocimetry (APTV) to transparent particles utilizing backlight illumination. Here, a particle acts as ball lens and bundles the light to a focal point, which is used to determine the particle's out-of-plane position. Due to the distance between focal point and particle, additional features have to be considered in ball lens astigmatism particle tracking velocimetry (BLAPTV) compared to conventional APTV. We describe required calibration steps and perform parameter studies to show how the autocorrelation coefficient and the light exposure affect the accuracy of the method. It is found that the accuracy and robustness of the Euclidean calibration approach as also used in conventional APTV (Cierpka et al. in Meas Sci Technol 22(1):015401, 2010a) can be increased if an additional calibration curve for the light intensity of the particle's focal point is considered. In addition, we study the influence of the particle diameter and the refractive index jump between liquid and particles on the calibration curves and the accuracy. In this way, particles of the same size, but different material, can be distinguished by their calibration curve. Furthermore, an approach is presented to account for shape changes of the calibration curve along the depth of the measurement volume. Overall, BLAPTV provides high out-of-plane particle reconstruction accuracies with respect to the particle diameter. In test cases, position uncertainties down to 1.8% of the particle diameter are achieved for particles of d p = 124 μm. The measurement technique is validated for a laminar flow in a straight rectangular channel with a cross-sectional area of 2.3 × 30 mm 2. Uncertainties of 0.75% for the in-plane and 2.29% for out-of-plane velocity with respect to the maximum streamwise velocity are achieved.
In the present study, we demonstrate for the first time how Astigmatism Particle Tracking Velocimetry (APTV) can be utilized to measure suspensions dynamics. Measurements were successfully performed in monodisperse, refractive index matched suspensions of up to a volume fraction of $$\varPhi =19.9\%$$ Φ = 19.9 % . For this, a small percentage ($$\varPhi <0.01\%$$ Φ < 0.01 % ) of the particles is labeled with fluorescent dye acting as tracers for the particle tracking procedure. Calibration results show, that a slight deviation of the refractive index of liquid and particles leads to a strong shape change of the calibration curve with respect to the unladen case. This effect becomes more severe along the channel height. To compensate the shape change of the calibration curves the interpolation technique developed by Brockmann et al. (Exp Fluids 61(2): 67, 2020) is adapted. Using this technique, the interpolation procedure is applied to suspensions with different volume fractions of $$\varPhi <0.01\%$$ Φ < 0.01 % , $$\varPhi =4.73\%$$ Φ = 4.73 % , $$\varPhi =9.04\%$$ Φ = 9.04 % , $$\varPhi =12.97\%$$ Φ = 12.97 % , $$\varPhi =16.58\%$$ Φ = 16.58 % and $$\varPhi =19.9\%$$ Φ = 19.9 % . To determine the effect of volume fraction on the performance of the method, the depth reconstruction error $$\sigma _z$$ σ z and the measurement volume depth $$\varDelta z$$ Δ z , obtained in different calibration measurements, are estimated. Here, a relative position reconstruction accuracy of $$\sigma _z$$ σ z /$$\varDelta z$$ Δ z = 0.90% and $$\sigma _z$$ σ z /$$\varDelta z$$ Δ z = 2.53% is achieved for labeled calibration particles in dilute ($$\varPhi <0.01\%$$ Φ < 0.01 % ) and semi-dilute ($$\varPhi \approx 19.9\%$$ Φ ≈ 19.9 % ) suspensions, respectively. The measurement technique is validated for a laminar flow in a straight rectangular channel with a cross-sectional area of 2.55 × 30 mm$$^2$$ 2 . Uncertainties of 1.39% and 3.34% for the in-plane and 9.04% and 22.57% for the out-of-plane velocity with respect to the maximum streamwise velocity are achieved, at solid volume fractions of $$\varPhi <0.01\%$$ Φ < 0.01 % and $$\varPhi =19.9\%$$ Φ = 19.9 % , respectively.
We present a methodology that allows to measure the dynamics of polydisperse suspension flows by means of Astigmatism Particle Tracking Velocimetry (APTV). Measurements are successfully performed with tridisperse suspensions flows in a square duct of up to $$\varPhi =9.1\%$$ Φ = 9.1 % particle volume fraction. Using a refractive index matching technique, a small amount of the particles ($$\varPhi =0.08\%$$ Φ = 0.08 % ) is labeled with fluorescent dye to be visible to the camera during the particle tracking procedure. Calibration measurements are performed for ten different particles diameters $$d_p$$ d p ranging from $$d_p= {15}\upmu \mathrm{m}$$ d p = 15 μ m to $$d_p= {260}\,\upmu \mathrm{m}$$ d p = 260 μ m . It is shown that Euclidean calibration curves of different $$d_p$$ d p overlap outside the focal planes, which induces ambiguities in a polydisperse APTV measurement. In the present approach, this ambiguity can be overcome utilizing the light intensity of a particle image which increases sharply with $$d_p$$ d p . In this way, extended Euclidean calibration curves can be generated for each particle group which are spatially separated through the light intensity which serves as an additional calibration parameter (Brockmann et al. in Exp Fluids 61(2):67, 2020). The extended Euclidean calibration allows to simultaneously differentiate particles of different sizes and determine their 3D location. This facilitates to investigate the migration behavior of mono- and tridisperse suspension flows which we demonstrate here for square duct flows with cross-sectional areas of $$0.6\times 0.6\,\mathrm{mm}^2$$ 0.6 × 0.6 mm 2 and $$0.4\times 0.4\,\mathrm{mm}^2$$ 0.4 × 0.4 mm 2 at bulk Reynolds numbers of $$\mathrm{Re}_b \approx 20$$ Re b ≈ 20 and $$\mathrm{Re}_b \approx 40$$ Re b ≈ 40 for particle volume fractions of $$\varPhi =0.08\%$$ Φ = 0.08 % and $$\varPhi =9.1\%$$ Φ = 9.1 % . At $$\varPhi =0.08\%$$ Φ = 0.08 % and $$\mathrm{Re}_b=20$$ Re b = 20 , we observe particles to arrange themselves in a ring-like formation inside the capillary, henceforth referred to as Pseudo Segré Silberberg Annulus (PSSA), with no significant differences between mono- and polydisperse suspension particle distributions. At $$\varPhi =9.1\%$$ Φ = 9.1 % , particles in monodisperse suspensions scatter around the PSSA. This scattering decreases when $$d_p$$ d p increases or $$Re_b$$ R e b increases from 20 to 40. Striking differences are observed in polydisperse suspensions. Large particles ($${60}\,\upmu \mathrm{m}$$ 60 μ m ) scatter significantly less around the PSSA in the polydisperse case compared to a monodisperse suspension of the same overall volume fraction. In contrast, small and intermediate particles ( $${30}\,\upmu \mathrm{m}$$ 30 μ m , $${40}\,\upmu \mathrm{m}$$ 40 μ m ) are repelled by larger particles resulting in regions of high concentration close to the channel walls which can be only observed in the polydisperse case. Graphical abstract
A dilute suspension in annular shear flow under gravity was simulated using multi-particle collision dynamics (MPC) and compared to experimental data. The focus of the analysis is the local particle velocity and density distribution under the influence of the rotational and gravitational forces. The results are further supported by a deterministic approximation of a single-particle trajectory and OpenFOAM CFD estimations of the overcritical frequency range. Good qualitative agreement is observed for single-particle trajectories between the statistical mean of MPC simulations and the deterministic approximation. Wall contact and detachment however occur earlier in the MPC simulation, which can be explained by the inherent thermal noise of the method. The multi-particle system is investigated at the point of highest particle accumulation that is found at 2/3 of the particle revolution, starting from the top of the annular gap. The combination of shear flow and a slowly rotating volumetric force leads to strong local accumulation in this section that increases the particle volume fraction from overall 0.7% to 4.7% at the outer boundary. MPC simulations and experimental observations agree well in terms of particle distribution and a close to linear velocity profile in radial direction.
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