Imaginary particle tracking accelerometry based on time-resolved velocity fields

Abstract

An accurate calculation of material acceleration is important for particle image velocimetry-based pressure reconstruction. Therefore, an imaginary particle tracking accelerometry (IPTA) approach based on time-resolved velocity fields is described in this paper for a better determination of acceleration. Multi-velocity fields and a least squares polynomial fitting of the velocity along imaginary particle trajectories are introduced to improve the acceleration accuracy. The process of imaginary particle tracking is operated iteratively until a convergence condition is satisfied. Then the Lagrangian acceleration (or the material acceleration in the Eulerian coordinates) is acquired by the first-order time derivation of the fitting polynomial. In addition, the sensitivity of the IPTA approach to different levels of noise and parameters that affect its performance is investigated. A criterion is proposed to determine these parameters when using IPTA to calculate the acceleration. Performance of the IPTA method is compared with other velocity-based accelerometry methods, including both Eulerian and Lagrangian methods. Assessments are conducted in a synthetic solid body rotation flow, a synthetic flow of a vortex ring, and an experimental jet flow. The results show that IPTA is a robust method for experimental acceleration determination that can both improve the accuracy of acceleration and provide better physical characteristics of the flow field.

Appendices

Appendix 1: Comparison between the IPTA method and the method by Pröbsting et al. (2013)

As mentioned in Sect. 3.1.4, the accelerometry method by Pröbsting et al. (2013) also applied a least squares estimator, but without any iteration process (denoted as “LSE” in this appendix). This method is not included in the test in Sect. 3.1.4 because their least squares procedure is different from that in the IPTA method, and thus cannot reflect whether it is the least squares polynomial fitting or the iteration procedure that improves the acceleration accuracy. As Sect. 3.1.4 has solved this question, it is now necessary to figure out the difference between the two kinds of least squares procedures.

As mentioned in the main context, the LSE method is based on an assumption of a constant acceleration over the whole evaluation time. This assumption makes the application of the method limited to the cases in which the evaluation time is not large and acceleration variation is small. Otherwise, the accuracy cannot be guaranteed, such as in the flow with Karman vortex street, where acceleration changes spatially and temporally. That is why the least square polynomial fitting is applied in the IPTA method. By setting the order of the polynomial for fitting velocity larger than 2, acceleration during the evaluation time is allowed to change. In this way, the accuracy of the calculated acceleration can be improved. To assess the capability of the two kinds of least square procedures when dealing with complicated flow, a velocity field is simulated. Trajectories of imaginary particles in the flow follow a sinusoidal curve. The trajectory of an imaginary particle started at (x, y) = (0, y ₀) can be expressed as

$$\left\{ \begin{aligned} x &= t \hfill \\ y &= y_{0} + \sin (\pi t). \hfill \\ \end{aligned} \right.$$

(12)

The theoretical velocity and acceleration can be calculated easily. Similar to the assessments in Sect. 3, spatially correlated Gaussian-distributed noise is generated and added into the velocity fields with the maximum velocity noise equal to 2% of the mean magnitude of velocity. For the IPTA method, the polynomial order for calculation is k = 3. Two velocity numbers are chosen as N = 5 and N = 21. The velocity numbers for the IPTA and LSE methods are maintained the same. The turnover time of the flow is selected as the cycle period in the y direction which is 2 in non-dimensional unit. Therefore, the total criterion for the IPTA method is 0.014 < Δt < 0.199 when N = 5, and 0.003 < Δt < 0.040 when N = 21. The Eulerian method and the pseudo-tracing method are also included in the assessment.

Figure 18 shows the total acceleration errors from different methods. Δt in (a) is five times Δt in (b). Therefore, Fig. 18a, b shares the same total evaluation time, which is similar to Fig. 13. Acceleration errors for the Eulerian and pseudo-tracing methods are not related to velocity numbers. The large errors when Δt is small for the two methods are due to the large uncertainty of using only two velocity fields. The minimum errors for the Eulerian and pseudo-tracing methods at different time steps are 6.5 and 3.7%, respectively. Errors of both the IPTA and the LSE methods are closely related to the time step Δt and total evaluation time (N − 1)Δt. For a certain velocity number of N, the LSE method is more accurate when Δt is small, because the assumption of constant acceleration over the whole evaluation time for the LSE method is reasonable under this circumstance. As a result, the random errors are better suppressed due to the embedded average smoothing effect in the LSE method. When Δt gets larger, the IPTA method tends to be more accurate, because the assumption of the constant acceleration becomes unacceptable, while an order of k = 3 for the polynomial fitting on the velocity in the IPTA method becomes superior. For a certain total evaluation time, both the methods tend to have better performance on increasing the sample frequency (by comparing Fig. 18a, b). The minimum errors for the LSE and IPTA methods at different time steps are 3.1 and 2.6% for N = 5, and 2.0 and 1.5% for N = 21, respectively. It indicates that with the procedures of iteration and a least square polynomial fitting, the IPTA method can achieve a more accurate result. Both the results are better than those obtained by the Eulerian and pseudo-tracing methods. Although both the LSE method and the IPTA method have large errors when Δt is small for a certain N, these two methods can improve the accuracy by increasing the velocity number of N, while the Eulerian and pseudo-tracing methods cannot utilize the information from the high-frequency sampling. Furthermore, the total criterion of Eq. (9) provides a way for the IPTA method to determine the value of N.

Fig. 18

Total errors from different methods when Δt changes. a N = 5, k = 3; b N = 21, k = 3

Appendix 2: IPTA pseudo-code