On the calibration of Astigmatism particle tracking velocimetry for suspensions of different volume fractions

Abstract

Abstract 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 $$\varPhi =19.9\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>19.9</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> . For this, a small percentage ( $$\varPhi &lt;0.01\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> ) 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 $$\varPhi &lt;0.01\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , $$\varPhi =4.73\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4.73</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , $$\varPhi =9.04\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>9.04</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , $$\varPhi =12.97\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>12.97</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , $$\varPhi =16.58\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>16.58</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> and $$\varPhi =19.9\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>19.9</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> . To determine the effect of volume fraction on the performance of the method, the depth reconstruction error $$\sigma _z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>z</mml:mi> </mml:msub> </mml:math> and the measurement volume depth $$\varDelta z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>z</mml:mi> </mml:mrow> </mml:math> , obtained in different calibration measurements, are estimated. Here, a relative position reconstruction accuracy of $$\sigma _z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>z</mml:mi> </mml:msub> </mml:math> / $$\varDelta z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>z</mml:mi> </mml:mrow> </mml:math> = 0.90% and $$\sigma _z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>z</mml:mi> </mml:msub> </mml:math> / $$\varDelta z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>z</mml:mi> </mml:mrow> </mml:math> = 2.53% is achieved for labeled calibration particles in dilute ( $$\varPhi &lt;0.01\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> ) and semi-dilute ( $$\varPhi \approx 19.9\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>19.9</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> ) 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 $$^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . 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 $$\varPhi &lt;0.01\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.01</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> and $$\varPhi =19.9\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>19.9</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , respectively.

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