# Coupling temporal and spatial gradient information in high-density unstructured Lagrangian measurements

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## Abstract

Particle tracking velocimetry (PTV) produces high-quality temporal information that is often neglected when computing spatial gradients. A method is presented here to utilize this temporal information in order to improve the estimation of spatial gradients for spatially unstructured Lagrangian data sets. Starting with an initial guess, this method penalizes any gradient estimate where the substantial derivative of vorticity along a pathline is not equal to the local vortex stretching/tilting. Furthermore, given an initial guess, this method can proceed on an individual pathline without any further reference to neighbouring pathlines. The equivalence of the substantial derivative and vortex stretching/tilting is based on the vorticity transport equation, where viscous diffusion is neglected. By minimizing the residual of the vorticity-transport equation, the proposed method is first tested to reduce error and noise on a synthetic Taylor–Green vortex field dissipating in time. Furthermore, when the proposed method is applied to high-density experimental data collected with ‘Shake-the-Box’ PTV, noise within the spatial gradients is significantly reduced. In the particular test case investigated here of an accelerating circular plate captured during a single run, the method acts to delineate the shear layer and vortex core, as well as resolve the Kelvin-Helmholtz instabilities, which were previously unidentifiable without the use of ensemble averaging. The proposed method shows promise for improving PTV measurements that require robust spatial gradients while retaining the unstructured Lagrangian perspective.

## 1 Introduction

The aerodynamic and hydrodynamic forces of highly-separated vortical flows are often analyzed with respect to the topology of the flow, for instance with Lagrangian coherent structures as demonstrated in Rockwood et al. (2016). PTV inherently provides this Lagrangian perspective on the development of the flow, as was illustrated above in Fig. 1. For instance, fluid that passes over a vorticity source like the leading-edge shear layer in Fig. 1 can be tracked forward along a pathline to study the evolution of that mass, in this case into a vortex. Meanwhile, the mass in this vortex can be tracked backwards in time to identify the source of this vorticity. Additionally, the high-quality substantial derivative determined from PTV dramatically improves the quality of pressure fields derived from the velocity field, as described by Neeteson and Rival (2015) and Neeteson et al. (2016). Wolf et al. (2013) also noted that conducting a Lagrangian-frame analysis alongside familiar Eulerian techniques yielded unique insight into vortex entrainment. However, most available high-resolution techniques for determining spatial gradients do so on the familiar Eulerian grid, while averaging techniques for Lagrangian data smooth away small-scale flow structures. Therefore, with the motivation of maintaining the intrinsic transport information of a pathline, the current study proposes a high-resolution method for determining derived properties natively on unstructured Lagrangian data, without resorting to averaging multiple runs.

In order to evaluate the robustness of the proposed methodology, it is applied to two test cases: First, a synthetic data set, for which true values of the velocity gradient tensor are known a priori, is used to verify that the method reduces gradient estimation error in the process of reducing noise. Second, an experimental 4D-PTV data set is used to evaluate the increase in fidelity and reduction in noise when the method is applied to real experimental data. The details of the vorticity-correction technique are presented next.

## 2 Background

By directly tracking individual particles, PTV does not experience the loss of fidelity that occurs when taking the average displacement across an interrogation window, as described by Kaehler et al. (2012). Furthermore, by collecting data along a pathline, high-quality temporal information is obtained, especially in the form of substantial derivatives. However, if one wishes to maintain the unstructured Lagrangian description of the flow, there are limited methods for computing spatial gradients. Among them there are regression-based methods that, for example, minimize the residual of an overdetermined set of directional derivatives, as discussed by Meyer et al. (2001). Alternatively, there are weighted-averaging techniques that calculate the gradient as a weighted-sum of local derivatives, as discussed by Correa et al. (2011).

*N*pathlines are enumerated by the counter

*p*, while each pathline is tabulated in the vector \(\mathbf {L}\), such that the number of frames in which a particle is observed is denoted \(\mathbf {L}(p)\). At each timestep along a pathline, the position \(\mathbf {X}\), velocity \(\mathbf {U}\) and acceleration \(\mathbf {a}\) of each particle are known for

*i*timesteps. We will denote the components of these vectors with

*j*.

*O*, given our initial guess. There are many ways to perform this optimization, but as a proof of concept we will use a simple gradient descent method. \(\nabla O\) is determined with respect to all elements of \(\mathbf {U}(p,i)\).

*O*is then minimized by iteratively adjusting each individual element of \(\nabla \mathbf {U}(p)\) along the direction \(\nabla O\) by a step \(\alpha \):

- 1.
An initial estimate of the velocity gradient tensor is produced, across the entire set of particles at each timestep, using an established method, such as from an overdetermined set of directional derivatives (see: Meyer et al. 2001), weighted-averaging technique (see: Correa et al. 2011), or interpolation.

- 2.
The data set is separated into pathlines, such that each pathline has \(9\times \mathbf {L}(p)\) velocity gradient components to be adjusted.

- 3.
These \(9\times \mathbf {L}(p)\) velocity gradient components are evaluated based on the objective function in Eq. (3).

- 4.
The gradient of this objective function is determined by making small adjustments to the \(9\times \mathbf {L}(p)\) velocity gradient components.

- 5.
New values of the velocity gradient components are determined using a steepest descent optimization following Eq. (4) until a local optimum of the objective function is found.

In the following section, the above methodology will be evaluated on a synthetic set of pathlines in order to have access to “true” reference values, and evaluate error reduction. Following this evaluation, the methodology will be implemented on experimental data to both evaluate noise reduction on real data, and to demonstrate Lagrangian analysis.

## 3 Numerical test case: dissipating Taylor–Green vortex

Since a physical measurement would not provide any insight into error reduction for the proposed methodology, the initial evaluation presented here utilizes a synthetic data set. A dissipating Taylor–Green vortex field in three dimensions is used for our test case (see Taylor and Green 1937). The data set for the Eulerian field was generated using a pseudo-spectral code whose formulation and validation is briefly described below. Simulated particles that perfectly follow the flow were advected through this domain, and Gaussian synthetic noise was added to the particle positions to simulate measurement error. Using the mean frame-to-frame displacement as a reference value, random displacements of between 0 and 3% were applied to each particle. This is comparable to the reconstruction quality of experimental techniques such as SMART (0.2 px error over 6 px frame-to-frame displacement; see Schanz et al. (2016). Particle positions were subsequently regularized using a five-snapshot, second-order polynomial fit. The error was subsequently evaluated both before and after applying the proposed methodology. The results for this synthetic test case described below, following a brief description of the computational method.

### 3.1 Computational methodology

Equations (5) and (6) are solved in spectral (Fourier) space using a pseudo-spectral method (see Orszag 1969, 1972). To remove the aliasing error, the 3/2 rule proposed by Orszag (1971) was adopted. Periodic boundary conditions are applied in all three directions with a domain size of \(2\pi \times 2\pi \times 2 \pi \). The governing equations are integrated in time using fractional step method (see Kim and Moin 1985) with a second-order, three-step Runge-Kutta time-advancement scheme.

*u*-component of velocity are shown in Fig. 3 for the current simulation and those of Brachet et al. (1983). Figure 3 (left) is the initial condition. Figure 3 (middle) is the current simulation at \(t = 5.0\), which is the same as the results of Brachet et al. (1983), presented in Fig. 3 (right). In addition to this qualitative comparison, the dissipation rate, \(\varepsilon = \nu \langle \frac{\partial u_i}{\partial x_j} \frac{\partial u_i}{\partial x_j} \rangle \) (where \(\left<...\right>\) indicates volume averaging), is also compared with literature and is shown in Fig. 4. The result of the current simulation is in very good agreement with that of Brachet et al. (1983).

### 3.2 Results: proposed method operating on the dissipating Taylor–Green vortex

*O*as expressed in Eq. (3), as a function of iteration. The function quickly converges upon its final value within five to ten iterations, with an overall reduction of approximately 60% versus the initial value of the function.

## 4 Experimental test case: starting vortex on an accelerating circular plate

Given the reduction of error produced by the vorticity-correction method on synthetic data, as demonstrated in Fig. 5, we will now apply the method to experimental data derived from a single image sequence. The test case consists of the vortex wake behind a circular flat plate towed at a constant acceleration. The experimental setup is explained below.

### 4.1 Experimental apparatus and image-processing methodology

*D*= 30 cm was accelerated normal to its path at a dimensionless acceleration of \(a^*=aD^3/\nu ^2=1.07\times 10^{10}\), where

*a*and \(\nu \) represent dimensional acceleration and kinematic viscosity, respectively. The motion was achieved by a rack-and-pinion traverse above the towing tank. The sting holding the circular plate was 2

*D*long, with a circular profile and a diameter of 0.1

*D*, attaching to the plate on its suction side. The blockage ratio of the experiment is \(7\%\). The sting assembly and optical setup are shown in Fig. 8b. Further documentation on this experiment can be found in Fernando and Rival (2016).

\(55 \ \upmu \rm{m}\) polymer spheres were seeded in the flow to serve as tracer particles. The Stokes number of the particles was approximately \(3\times 10^{-3}\), which ensured tracer-accuracy errors of \(<1\%\) (see Raffel et al. 2007). The tracers were illuminated by a \(527~\text {nm}\), 40mJ-per-pulse laser expanded into a \(10\times 10\times 0.3~\)cm\({}^3\) volume. Four Photron SA4 high-speed cameras captured images of the tracers within this volume at a frame rate of 900 \(\text {Hz}\). To minimize image distortions, water-filled prisms were fixed onto the glass pane of the tank such that all cameras were orthogonal to a prism face. The acquired images were then processed in DaVis 8.3.0 and using a 4D-PTV tracking algorithm; see Schanz et al. (2016) for details. Measurements were performed over a diameters-traveled domain of \(0.1\le s/D \le 0.28\), which corresponds to circulation-based Reynolds numbers between \(9\times 10^3\le {Re}_\Gamma \le 43\times 10^3\). Finally, the 4D-PTV pathlines were then extended forwards and backwards beyond their original lifespan via a pathline-extension method inspired by flow-map compilation techniques described in Brunton and Rowley (2010) and Raben et al. (2014).

### 4.2 High-fidelity measurements of the starting vortex growth

*before*image is of the initial velocity gradient estimate produced utilizing the method of Meyer et al. (2001). The noise of the starting vortex is reduced, allowing for the identification of structures not previously obvious, such as individual Kelvin-Helmholtz instabilities in the shear layer. These structures would also be obscured by ensemble-averaging, such that the ability to correct individual runs is critical. This noise reduction is more clearly shown in the supplementary video “circular_flat_plate.mp4”. The concentric circular vorticity levels also clearly identify the vortex core in the corrected vorticity field. Although the vorticity scale is saturated in Fig. 8a, it is worth noting that the small-scale structures remain obscured at all scaling levels. Furthermore, it should be noted that the vorticity correction method will often increase velocity gradient magnitudes, and does not simply smooth the gradient values. The computational cost of the proposed methodology did not significantly increase for the experimental data relative to the synthetic data, correcting pathlines on the order of 10 per second on a desktop-class computer.

*O*as expressed in Eq. (3) is shown in Fig. 10. Despite the simple optimization method used in this study, the residual of most pathlines converged on its final value after five to ten iterations. However, it is worth noting that in the absence of an objective true value with which to compare to, no quantitative treatment of error reduction can be given here. The convergence on a local minimum, as opposed to zero, was seen as acceptable in order to avoid gradient estimations tending towards zero.

## 5 Conclusions

The purpose of this study was to improve the estimation of spatial gradients on unstructured Lagrangian data without discarding any pathline information. Such robust gradient estimation can improve the understanding of vorticity transport through a flow for the purposes of aerodynamic or hydrodynamic optimization or flow control. Therefore, we proposed a gradient correction scheme based on the knowledge that the substantial derivative of vorticity through a flow must equal the vortex stretching/tilting through that flow. This constraint was realized by minimizing the residual of the vorticity-transport equation across all points of a pathline simultaneously.

The proposed method has been implemented on both synthetic and experimental data consisting of a decaying Taylor–Green vortex field and an accelerating circular plate, respectively. In the synthetic case, mean errors were shown to be consistently reduced by the proposed vorticity-correction method across all seeding densities, by up to 40%. However, as the method shown here is locally optimizing, the error reduction achieved is dependent on the spatial resolution of the initial gradient estimate. Meanwhile, the application of the proposed method to experimental data reduced the vorticity-transport residual by approximately 40%. The proposed method also provided access to small-scale flow structures such as the Kelvin-Helmholtz instabilities that were otherwise obscured. This retention of Lagrangian data was demonstrated with the direct investigation of material transport within a starting vortex. Small-scale flow structures could be identified in the post-processed data that were otherwise unavailable to the raw gradient outputs. By identifying those flow structures one could then track the vorticity-containing mass back to its origin. This experimental case demonstrates the value of retaining Lagrangian data for aerodynamic or hydrodynamic optimization, which could eventually lead to new insights when studying complex, vortical flows.

## Supplementary material

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