14th International Symposium on Particle Image Velocimetry – ISPIV 2021
August 1–5, 2021

A grid-free least-squares method for pressure
evaluation from LPT data

M. Bobrov1,2, M. Hrebtov1,2, V. Ivashchenko1, R. Mullyadzhanov1,2,
A. Seredkin1,2, M. Tokarev1,2, D. Zaripov1,3,∗, V. Dulin1,2, D. Markovich1,2

1 Institute of Thermophysics SB RAS, Lavrentyev ave. 1, 630090 Novosibirsk, Russia
2 Novosibirsk State University, Pirogov str. 2, 630090 Novosibirsk, Russia

3 School of Mechanical Science and Engineering, Huazhong University of Science and Technology,
430074 Luoyu Road 1037, Wuhan, China

∗zaripov.d.i@gmail.com

Lagrangian particle tracking Shake-the-box (STB) method (Schanz et al., 2016) acquires the 3D po-
sitions of tracer particles from the temporal sequences of their 2D projection images even for rather high
seeding densities. Approximation of tracks by analytical functions (Gesemann, 2015) provides an accurate
evaluation of tracers’ local velocity and acceleration. This data, which is obtained on non-regular grid, can
be used to estimate local pressure fluctuations based on the Navier–Stokes equation. The present paper
describes a grid-free least-squares method for the gradients and pressure evaluation based on irregularly
scattered LPT data with random noise minimization.

The performance of the employed method is assessed based on synthetic images of virtual particles
seeded in a wall-bounded turbulent flow. The tracks are obtained from the direct numerical simulation
(DNS) of an initially laminar boundary layer flow around a hemi-sphere, mounted onto a flat wall. The
Reynolds number based on the sphere diameter and free stream velocity is 7000 corresponding to a fully
turbulent wake (see Figure 1).

Figure 1: Location of the virtual cameras with respect to the coordinate system and a flow domain where
particles were initially seeded (larger box) and ROI (smaller box) with visualization of the vortex structures
with instantaneous isosurface of λ2 =−5 colored by streamwise velocity magnitude

In order to study the effect of particle image concentration (up to 0.2 ppp) on the overall accuracy, two
hundred thousand inertial particles were randomly and uniformly seeded into a flow domain as illustrated



in Figure 1. We considered four concentrations, i.e. Nppp = 0.005, 0.05, 0.12 and 0.2 ppp. To quantify
the image noise effect, a random noise with different amplitude was added to the projection images for the
moderate case of Nppp = 0.05. The images were processed using an STB algorithm (similar to that of Gese-
mann) enabling an accurate tracking of particles and evaluation of their velocity and acceleration. The first-
and second-order gradients of the velocity and acceleration were evaluated using a grid-free least-squares
method (Mavriplis, 2003). Afterwards the local values of pressure were computed employing a modified
version of the method, where the pressure gradients were obtained directly from the Navier-Stokes equations
(Kuhnert and Tiwari, 2004). Besides, in order to reveal the effect of spatial resolution and performance of
the employed algorithms, the original tracks from the DNS were used to evaluate the same flow quantities.
We refer to such case as Hacker’s data.

The examples of the results are presented in Figures 2 and 3. Particle image concentration Nppp = 0.05
is shown in Figure 2 for the comparison with the Hacker’s data. The velocity and acceleration field are
interpolated to the regular grid using the least-squares method, whereas the pressure is evaluated from the
Poison equation. The resolution is sufficient to recover large-scale flow structures and negative pressure
regions in the wake.

Figure 2: The instantaneous distributions of the wall-normal velocity (left) and acceleration (middle) com-
ponents and pressure (right) obtained for Hacker’s and STB data

Figure 3: The root-mean-square deviations of the error between the exact and estimated velocities, velocity
gradients, accelerations and pressure (from left to right) for different concentrations and image noise level
for Nppp = 0.05

The performance of the used STB algorithm is limited for the cases of high seeding density Nppp = 0.12
and noise level of 10%. Besides, the spatial resolution is not enough to approximate accurately the velocity
gradients and particles acceleration on the regular grid. However, the least-squares method demonstrates a



remarkable accuracy for the pressure evaluation from the exact particles’ locations, velocities and accelera-
tion (see Hacker’s data in Figure 3), viz., even for Nppp = 0.05 the pressure error is below 10%.

Acknowledgements
This work was financially supported by the Russian Science Foundation, grant number 19-79-30075.

References
Gesemann S (2015) From particle tracks to velocity and acceleration fields using b-splines and penalties.

arXiv preprint arXiv:151009034

Kuhnert J and Tiwari S (2004) Grid free method for solving the poisson equation. Wavelet Analysis and
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Schanz D, Gesemann S, and Schröder A (2016) Shake-the-box: Lagrangian particle tracking at high particle
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