Three-dimensional time-resolved Lagrangian flow field reconstruction based on constrained least squares and stable radial basis function

Abstract

The three-dimensional time-resolved Lagrangian particle tracking (3D TR-LPT) technique has recently advanced flow diagnostics by providing high spatiotemporal resolution measurements under the Lagrangian framework. To fully exploit its potential, accurate and robust data processing algorithms are needed. These algorithms are responsible for reconstructing particle trajectories, velocities, and differential quantities (e.g., pressure gradients, strain- and rotation-rate tensors, and coherent structures) from raw LPT data. In this paper, we propose a 3D divergence-free Lagrangian reconstruction method, where three foundation algorithms—constrained least squares (CLS), stable radial basis function (RBF-QR), and partition-of-unity method (PUM)—are integrated into one comprehensive reconstruction strategy. Our method, named CLS-RBF PUM, is able to (1) directly reconstruct flow fields at scattered data points, avoiding Lagrangian-to-Eulerian data conversions; (2) assimilate the flow diagnostics in Lagrangian and Eulerian descriptions to achieve high-accuracy reconstruction; (3) process large-scale LPT datasets with more than hundreds of thousand particles in two dimensions (2D) or 3D; (4) enable inter-frame and inter-particle interpolation while imposing physical constraints (e.g., divergence-free for incompressible flows) at arbitrary time and locations. Validation based on synthetic and experimental LPT data confirm that our method can achieve the above advantages with accuracy and robustness.

Graphical abstract

Appendices

Appendices

A Differential quantity

1.1 A.1 Strain-rate and rotation-rate tensor

Strain-rate and rotation-rate tensors are kinematics quantities that describe the rate of change of a fluid parcel regarding deformation and rotation, respectively. The strain-rate tensor ${{\varvec{S}}}$ and rotation-rate tensor ${{\varvec{R}}}$ are the symmetric and anti-symmetric parts of the velocity gradient $\nabla {{\textbf {U}}}$, respectively: $\nabla {{\textbf {U}}} =\frac{1}{2}( U _{i,j}+ U _{j,i})+\frac{1}{2}( U _{i,j}- U _{j,i})$, ${{\varvec{S}}} = S_{ij} =\frac{1}{2}( U _{i,j}+ U _{j,i})$, and ${{\varvec{R}}} = R_{ij} =\frac{1}{2}( U _{i,j}- U _{j,i})$. Once all components of $U _{i,j}$ are calculated in Step 4, ${{\varvec{S}}}$ and ${{\varvec{R}}}$ can be evaluated.

1.2 A.2 Coherent structure based on Q-criterion

The Q-criterion is a vortex identification method first proposed by Hunt et al. (1988) to identify vortical structures in the flow. The Q-criterion is given by (Hunt et al. 1988; Haller 2005):

$$\begin{aligned} Q=\frac{1}{2}(\Vert {{\varvec{R}}}^2 \Vert - \Vert {\varvec{S}}^2 \Vert )>0 \end{aligned}$$

(15)

where ${\varvec{R}}$ is the rotation-rate tensor, and ${\varvec{S}}$ is the strain-rate tensor. After ${{\varvec{S}}}$ and ${{\varvec{R}}}$ are calculated, vortices can be found using Eq. (15). Other criteria can be achieved similarly.

B Synthetic and experimental data

1.1 B.1 Taylor–Green vortex (TGV)

To generate the TGV synthetic data for validations, we adopted a velocity field:

$$\begin{aligned} \begin{aligned} u&= \alpha _1 \cos {(\omega (x-d_x))}\sin {(\omega (y-d_y))}\sin {(\omega (z-d_z))} \\ v&= \alpha _2 \sin {(\omega (x-d_x))}\cos {(\omega (y-d_y))}\sin {(\omega (z-d_z))} \\ w&= \alpha _3 \sin {(\omega (x-d_x))}\sin {(\omega (y-d_y))}\cos {(\omega (z-d_z))} \end{aligned}, \end{aligned}$$

(16)

where $\alpha _1=\alpha _2=0.5$, $\alpha _3=-1$, and $\omega =2 \pi$; $d_x=d_y=d_z=0.25$ were the offsets of the TGV flow structure in the x, y and z directions, respectively. In the first snapshot, 20,000 particles were randomly placed in the domain $\Omega$ of $x \times y \times z \in [0,1]\times [0,1]\times [0,1]$. The particle locations in the next snapshot were calculated by the forward Euler method: ${\varvec{x}}_{\kappa +1}={\varvec{x}}_{\kappa }+\delta t {{\textbf {U}}}_{\kappa }$, where ${\varvec{x}}_{\kappa }=({x}_{\kappa },{y}_{\kappa },{z}_{\kappa })$ and ${\varvec{x}}_{\kappa +1}$ were the particle locations in the $\kappa$-th and $(\kappa +1)$-th snapshots, respectively; ${{\textbf {U}}}_{\kappa }=(u_{\kappa },v_{\kappa },w_{\kappa })$ was the velocity calculated by Eq. (16) in the $\kappa$-th snapshot; $\delta t$ was the time interval between snapshots and here we chose $\delta t=5 \times 10^{-5}$. After the integration, particle trajectories with $2 \times 10^{4}$ snapshots were down-sampled to 11 frames² to generate the ground truth of the raw LPT data. Artificial noise (see Sect. 4 for details) was added to the particle spatial coordinates of the TGV-based ground truth.

1.2 B.2 Turbulence wake flow (TWF)

To generate the synthetic TWF data, we added artificial noise to an available Lagrangian dataset. Details about the dataset can be found in Khojasteh et al. (2022) and we briefly summarize it here. The dataset are created based on a DNS. The DNS was conducted using the Incompact3D solver (Laizet and Lamballais 2009) to simulate a flow with a free stream velocity U passing a cylinder with a diameter D at a Reynolds number of $Re=3,900$.

In the Lagrangian datasets, about 200,000 particles were scattered in the domain. The particle velocities were calculated by tri-linear interpolations using the nearest eight nodal data in space. The particle trajectories were integrated by a fourth-order Runge–Kutta method. The synthetic data had 105,000 particles that were down-sampled from the Lagrangian dataset. These particles were distributed in a wake region, located 0.5D downstream of the cylinder, with the dimensions of $x \times y \times z \in [6D,8D]\times [3D,5D]\times [2D,4D]$. The center-line of the cylinder located at $(x,y)=(5D,4D)$ is parallel to the z axis. A total of 11 frames were selected from 350 DNS frames and these 11 frames had a time interval $\Delta t=0.00375U/D$. The noise was defined the same way as that used in the TGV validation, with $\zeta =0.1\%$ to represent a typical noise in LPT data.

1.3 B.3 Low-speed pulsing jet

The experimental facility consisted of a hexagonal water tank, a cylindrical piston, an impingement plate, and optical equipment. The synthetic pulsing jet was generated by driving the piston using an electromagnetic shaker, and pushing the water through a circular orifice until impinged on the plate. A dual cavity high-speed laser illuminated the measurement area with the dimensions of $60 ~\text {mm} \times 57 ~\text {mm} \times 20~ \text {mm}$. Four high-speed cameras recorded the jet flow and provided time-resolved LPT images (see Sakib (2022) for details).

The raw experimental data were acquired using a volumetric system and commercial software LaVision DaVis 10 (Göttingen, Germany). The LPT module of DaVis 10 is based on the shake-the-box (Schanz et al. 2016) algorithm. The velocity data were post-processed by Vortex-In-Cell# (VIC#) algorithm (Jeon et al. 2022) to obtain velocity gradients on a structured mesh.

C Baseline algorithms

1.1 C.1 Finite difference methods

We use the velocity component u calculation as an example. In a given frame, the first- and second-order finite difference methods (FDMs) calculate velocities by the forward Euler method (i.e., Eq. 17a) and central difference method (i.e., Eq. 17b), respectively:

$$\begin{aligned} \tilde{u}(t_\kappa )&= \frac{\hat{x}_{\kappa +1}-\hat{x}_{\kappa }}{\Delta {t}}, \end{aligned}$$

(17a)

$$\begin{aligned} \tilde{u}(t_\kappa )&= \frac{\hat{x}_{\kappa +1}-\hat{x}_{\kappa -1}}{2\Delta {t}}, \end{aligned}$$

(17b)

where $\hat{x}_{\kappa }$ is the particle coordinate along the x direction in the $\kappa$-th frame from LPT experiments, $\Delta {t}$ is the time interval between two consecutive frames, and $t_\kappa$ is the time instant in the $\kappa$-th frame. The velocities of the first and last frames are calculated by the first-order finite difference method. Since the FDMs only evaluate particle velocities, the particle locations that are ‘output’ from the FDMs are assumed to be the same as those in the synthetic data, e.g., $\tilde{x}(t_\kappa )=\hat{x}(t_\kappa )$.

1.2 C.2 Polynomial regression

We use trajectory and velocity reconstruction in the x direction as an example. First, the trajectory polynomial model function $\tilde{x}(t)$ is given by $\tilde{x}(t)=\sum ^m_{j=0}{p_{m,j} t^j}$ where $p_{m,j}$ is the polynomial coefficient, m is the order of polynomials, and t is the time. For instance, the trajectory and velocity model functions of the 2nd polynomial (2nd POLY) are

$$\begin{aligned} \begin{aligned} \tilde{x}(t)=p_{2,2} t^2+ p_{2,1}t+p_{2,0} \\ \tilde{u}(t)=\dot{\tilde{x}}(t)=2p_{2,2} t+ p_{2,1} \end{aligned}. \end{aligned}$$

The trajectory and velocity approximation functions of the other polynomials are similar to those in the x direction of the 2nd POLY.

Next, we calculate the polynomial coefficient $p_{m,j}$. A residual $\mathcal {R}$ between measured particle locations $\hat{x}_{\kappa }$ and the polynomial model function $\tilde{x}({t}_{\kappa })$ is minimized $\min ~ \mathcal {R}=\sum ^{N_{\text {trj}}}_{\kappa =1}\Vert \tilde{x}({t}_{\kappa })-\hat{x}_{\kappa } \Vert ^2,$ where ${t}_{\kappa }$ is the time instant in the $\kappa$-th frame. Setting the gradient of the residual $\mathcal {R}$ regarding $p_{m,j}$ to zero (i.e., $\partial {\mathcal {R}}/\partial {p_{m,j}} =0$) and the polynomial coefficient $p_{m,j}$ can be explicitly solved. The polynomial coefficients in the y and z directions are the same as that in the x direction. Once all the coefficients are retrieved, the polynomial model functions can be constructed and used to approximate particle trajectories, velocities, and accelerations.

1.3 C.3 Cubic B-splines

The cubic B-spline algorithm is based on the MATLAB built-in functions from the Curve Fitting Toolbox™. We adopt a function spap2(piece,k,x,y) with piece=2 and k=4 to create a cubic spline with two pieces joined together, bypass setting knots. x and y are the given data points and their values, respectively. Then we use a function fnval(f,xe), in which f=spap2(piece,k,xe,y) is the spline function calculated above, and xe is the evaluation points. To calculate first-order derivatives for velocity as an example, we use fnder(f,d), where d=1.

D Spatial and temporal frequency response

We tested the spatial and temporal frequency response of our method using TGV and HIT. The TGV has varying predefined frequencies based on its analytical expressions (i.e., Eq. 16). It is used for both spatial and temporal frequency response analysis. The Lagrangian data based on the HIT have intrinsically varying frequencies along particle trajectories and we use them to test the temporal response.

For the temporal analysis based on the TGV, the velocity amplitudes $\alpha _1$, $\alpha _2$, and $\alpha _3$ in Eq. (16) are varied from 2 to 32 with an increment of 2, corresponding to the non-dimensional temporal frequency $f'_t$ of the flow varying from 1 to 16. Physically, $f'_t$ means the number of periodic circulations one particle in the outer layer of the vortex cell can finish within a unit of time. These TGV data consist of 33 frames with a time interval $\Delta t = 1/32$. For the spatial analysis based on the TGV, $\omega$ in Eq. (16) is varied from $\pi$ to $16 \pi$ with an increment of $\pi$, in turn, the non-dimensional spatial frequency $f'_s$ of the flow varies from 0.5 to 8. The frequency response is defined as the ratio of the amplitudes of the reconstructed particle spatial coordinates for temporal analysis, or of the reconstructed particle velocities for spatial analysis, to that of the input data in the frequency space. The frequency spectra are calculated by the fast Fourier transform. The normalized frequency $f^{*}$ is the spatial or temporal frequency of the flow divided by the Nyquist sampling frequency of the data. The parameters used in the frequency response analysis are listed in Table 2.

The TGV-based frequency response is presented in Fig. 12a and b. With increasing $f^{*}_t$ or $f^{*}_s$, the frequency response decreases from unity to approximately zero, with a cutoff frequency varying from about 0.2 to 0.4 for $f_{s}^*$ and $f_{t}^*$. This indicates a low-pass filter feature of our method. When the oversampling ratio for space ($\beta _1$ and $\beta _2$) and time ($\beta _0$) decreases, the cutoff frequency increases. This observation is expected, as using small oversampling ratios recovers high-frequency components while applying larger ones provides smooth reconstruction.

For the frequency response based on the HIT flow, we use synthetic data with artificial noise to examine how our method attenuates noise of different levels. The synthetic HIT data are generated by adding the artificial noise to the ground truth. Three artificial noise levels $\zeta =0.1\%$, $\zeta =1.0\%$, and $\zeta =5.0\%$ are used and they are defined the same as that in the validation (see Sect. 4). The ground truth is from a DNS of an incompressible flow with a Reynolds number $Re_{\lambda } \approx 130$ based on the Taylor scale (details about the HIT can be found in Biferale et al. (2023)). The non-dimensional frequency of this HIT flow is $f'_t=U_{\text {rms}}/L=0.275$, where $U_{\text {rms}}$ is the root mean squares of the magnitudes of the particle velocities in all frames; L is the length of the cubic computational domain. The $f'_t$ can be considered a metric describing the characteristic spatial frequency of the particle trajectory in this flow. The synthetic HIT data are sampled to 65 frames with a time interval $\Delta t = 0.00225$ between two consecutive frames. Therefore, the corresponding temporal frequency f based on sampling varies from 0 to about 215.28, with an increment of about 6.94. We assess the frequency response that is defined as the ratio of the amplitudes of the reconstructed particle spatial coordinates to that of the input noisy data in the frequency space.

The temporal frequency response using the HIT is shown in Fig. 12c. As illustrated in the figure, when the noise is high (e.g., $\zeta =5.0\%$), the high-frequency components (mainly noise) in the input data are sufficiently suppressed, while the flow frequency ones ($f^*_t<0.2$, which is dominated by true flow structures) are preserved. When the noise level decreases (e.g., $\zeta =1.0\%$ and $0.1\%$), our method recovers the particle trajectories at higher frequencies.

Table 2 Parameters for frequency response analysis

Fig. 12

The temporal and spatial frequency response of the CLS-RBF PUM method. Temporal (a) and spatial (b) frequency response based on the TGV data; c temporal frequency response based on the synthetic HIT data. c blue, green, and scarlet lines represent the frequency response based on the HIT data with 0.1%, 1.0%, and 5.0% noise, respectively