15th International Symposium on Particle Image Velocimetry – ISPIV 2023
June 19–21, 2023, San Diego, California, USA

Instantaneous 3D Flow Field Reconstruction with
Physics Informed Radial Basis Functions
P. Sperotto1∗ , P. Petersson1 , Bo Watz1 , D. Hess1
1 Dantec Dynamics, Copenhagen, Denmark.
∗ pietro.sperotto@dantecdynamics.com

Abstract
In this paper, we propose a method for approximating velocities and their gradients from 3D Particle
Tracking Velocimetry (PTV) data, using compactly supported Radial Basis Functions (RBFs), which enable
the treatment of large datasets.
The effectiveness of the proposed method is proven by using a synthetic test case, created by undersampling a simulated channel flow, with 𝑅𝑒𝑇 ≃ 1000,from the John Hopkins Turbulence Database (JHTD).
These synthetic measurements enable a fair comparison between the Data Assimilation (DA)/interpolation
algorithm and the ground truth. Three different algorithms are compared in this test case, the proposed
PI-RBFs, the classic RBF fitting and linear interpolation.
Additionally, an experiment is carried out to validate the applicability of the PI-RBF on the experimental
data. A simple convective flow is measured using a Tomographic PTV approach in a volume of interest
corresponding roughly to 30 × 16 × 11[𝑐𝑚 3 ] . The investigation of this large volume is made possible by theuse of Helium Filled Soap Bubbles (HFSBs) as seeding density. The size of these particles is roughly 300𝜇𝑚
giving a particle image size of 2 − 3 pixels.
The results based on the PI-RBFs show a significant improvement compared to linear and standard RBF
fitting. The enhancement becomes even more evident when derivatives of the velocities are analysed. All
methods show more or less the same main structures for the JHTD. But the PI-RBFs approach achieves
higher levels of detail coupled with increased smoothness of the gradients. This improvement is achieved
with a negligible increase in the computational cost, and thereby proves to be a valuable tool for meshless
DA approaches.

1

Introduction

In the last years, the increased computational power and improved algorithms such as Shake-The-Box (STB)
Schanz et al. (2016) permit to track higher particle densities without major post-processing, oppositely
to Particle Image Velocimetry (PIV). The main drawback of Particle Tracking Velocimetry (PTV) based
techniques is the scattered nature of the measured data. These techniques require interpolation/fitting toevaluate velocities into a regular grid and estimate gradients. Furthermore, differentiating experimental data
is a complex matter and it can lead to high errors, especially if no smoothing is applied.
To address these challenges, DA algorithms have been developed over the years such as Flowfit Gesemann(2015) and Vortex in cell (VIC+) Schneiders and Scarano (2016). Flowfit aims to find an approximation of
the flow field using B-splines; the algorithm does not limit finding the optimal fit of the PTV data but also
enforces in its fitting the flow governing equations. Instead, VIC+ solves the vorticity equations on a gridand optimizes the solution to fit the data iteratively. Moreover, Machine Learning can yield similar results.
For instance, recently in Zhou et al. (2022) Physics Informed Neural Network (PINN) was used to reduce
the errors in PTV measurements.
DA algorithms are different from standard interpolation approaches, which do not use physics to improve
the quality of their results. However, they are very popular due to their long-lasting heritage and low
computational costs. Among these interpolators and in particular Radial Basis Functions (RBFs) were used
often in literature (see Casa and Krueger (2013), Ratz et al. (2022) and Li et al. (2022)) due to their flexibility
and easy implementations in higher dimensions.
In Sperotto et al. (2022) RBFs were used with a data assimilation approach; the method allows to include
constraints such as no-slip condition and the divergence-free constraint into the fitting. However, this methodsuffered from memory management problems dictated by the globally supported RBFs (Gaussians), which
create full matrices. In the present contribution, we revisited this approach to manage larger datasets byimplementing Wendland RBF Wendland (1995). The resulting implementation allows for the computationof velocities with increased smoothness and quality requiring only to solve a linear system; the approach ismeshless and thus suitable for many applications. Solid obstacles in the measurement field does for example
not present a problem.
This paper will present in 2 the numerical approach. In 3 the test case will be explained and in 4 therelative results are shown. Finally, the conclusions and outlooks are proposed in 5.

2

Numerical approach

A RBF is defined as 𝜑 (∥𝒙 − 𝒙 𝑖 ∥ 2 , 𝑐) = 𝜑𝑖 (𝑟) with 𝑟 = ∥𝒙 − 𝒙 𝑖 ∥ 2 /𝑐 where 𝒙 ∈ R3 is a generic point in thespace, 𝒙 𝑖 ∈ R3 are the collocation points and 𝑐 ∈ R is the shapeÍfactor. A generic function function 𝑓 can𝑛𝑏= 𝜑𝑖 (𝑟) · 𝑤 𝑖 , where 𝑤 𝑖 are the weights
approximated by a linear combination of RBFs 𝑓 (𝒙) ≃ 𝑓˜ (𝒙) = 𝑖=0and 𝑛𝑏 is the total number of bases. In this work, the bases chosen are Wendland functions ∈ 𝐶 4 R3 (seeWendland (1995)), because of their compact support and smoothness.
We speak about RBF fits when the weights are sought to minimize the following cost function:
𝐽 (𝒘) = ∥ 𝒇 − ˜𝒇 ∥ 2(1)

where the bold symbol indicate that the function is now evaluated on a discrete set of points
𝒇 = ( 𝑓 (𝒙 0 ) , . . . , 𝑓 (𝒙 𝑘 ) , . . . , 𝑓 (𝒙 𝑁 )) ∈ R 𝑁 with 𝑁 the number of measured points and 𝒘 = 𝑤 0 , . . . , 𝑤 𝑖 , . . . , 𝑤 𝑛𝑏 ∈R𝑛𝑏 . Equation 1 can be compactly written as:
𝐽 (𝒘) = ∥ 𝒇 − 𝝓 · 𝒘∥ 22 + 𝛼𝑇 ∥𝒘∥ 22 ,(2)

where :

𝜑0 (𝑟 0 ) . . . 𝜑𝑖 (𝑟 0 ) . . . 𝜑 𝑛𝑏 (𝑟 0 )
ª
©
..
..
..
..
..
­
®
.
.
.
.
.
®
­
(3)
𝝓 = ­­ 𝜑0 (𝑟 𝑘 ) . . . 𝜑𝑖 (𝑟 𝑘 ) . . . 𝜑 𝑛𝑏 (𝑟 𝑘 ) ®® ∈ R 𝑁 ×𝑛𝑏 ,
..
..
..
­
®
..
..
.
.
­
®
.
.
.
(𝑟
)
(𝑟
)
(𝑟
)
𝜑
.
.
.
𝜑
.
.
.
𝜑
0
𝑁
𝑖
𝑁
𝑛
𝑁
𝑏
«
¬
and 𝛼𝑇 is the Tikhonov regularization that can be introduced to increase convexity of the cost function andpromote smoother solutions.
It can be shown that the 𝒘 , that minimize this cost function can be computed as: −1· 𝝓𝑇 · 𝒇 .
𝒘 = 𝝓𝑇 · 𝝓 + 𝛼𝑇 𝑰

(4)This formulation has been used effectively in literature, for instance, in Ratz et al. (2022).
We speak about Physics Informed RBF (PI-RBF) when the cost function changes to:  2
𝐽 (𝒘; 𝒙 L ) = ∥ 𝒇 − ˜𝒇 ∥ 22 + 𝛼 L L ˜𝒇(5)2

where L is a differential operator that represents the governing equation of the phenomena, 𝒙 L are the points
where the differential condition is enforced and 𝛼 L ∈ R is the penalty factor. The focus would be on thedivergence-free as L . In this situation equation 5 change to:∥𝒖 𝒙 − 𝒖˜ 𝒙 ∥ 2 ∥𝒖 𝒚 − 𝒖˜ 𝒚 ∥ 2 ∥𝒖 𝒛 − 𝒖˜ 𝒛 ∥ 2+++ 𝛼∇ ∇ ·𝐽 (𝒘; 𝒙 ∇ ) =∥𝒖 𝒙 ∥ 22∥𝒖 𝒚 ∥ 22∥𝒖 𝒛 ∥ 22

𝒖˜ 𝒙
𝒖˜ 𝒚
𝒖˜ 𝒛

! 2
,

(6)

2

for a complete description of the methodology and the demonstration of the resulting (positive definite) linear
system, the reader is referred to Sperotto et al. (2022).

Equation 5 can be further exploited to calculate the specific pressure. Indeed, once the velocities havebeen reconstructed it is possible to evaluate the gradient and Laplacian of the pressure in an arbitrary set ofpoints 𝒙 𝒑 ∈ R 𝑁 𝑝 :(7)𝜕𝑥 𝒑 𝑝 = − 𝜕𝑡 𝒖˜ 𝒙𝑝 + 𝒖˜ 𝒙𝑝 𝜕𝑥 𝒖˜ 𝒙𝑝 + 𝒖˜ 𝒚𝑝 𝜕𝑦 𝒖˜ 𝒙𝑝 + 𝒖˜ 𝒛𝑝 𝜕𝑧 𝒖˜ 𝒙𝑝 + 𝜈Δ𝒖˜ 𝒙𝑝 ;
𝑝
𝑝
𝑝
𝑝
𝑝
𝑝
𝑝
𝑝
𝑝
𝜕𝑦 𝒑 = − 𝜕𝑡 𝒖˜ 𝒚 + 𝒖˜ 𝒙 𝜕𝑥 𝒖˜ 𝒚 + 𝒖˜ 𝒚 𝜕𝑦 𝒖˜ 𝒚 + 𝒖˜ 𝒛 𝜕𝑧 𝒖˜ 𝒚 + 𝜈Δ𝒖˜ 𝒚 ;
(8)
𝑝
𝑝
𝑝
𝑝
𝑝
𝑝
𝑝
𝑝
𝑝
(9)
𝜕𝑧 𝒑 = − 𝜕𝑡 𝒖˜ 𝒛 + 𝒖˜ 𝒙 𝜕𝑥 𝒖˜ 𝒛 + 𝒖˜ 𝒚 𝜕𝑦 𝒖˜ 𝒛 + 𝒖˜ 𝒛 𝜕𝑧 𝒖˜ 𝒛 + 𝜈Δ𝒖˜ 𝒛 ;





2
2
2
Δ 𝒑 𝑝 = − 𝜕𝑥 𝒖˜ 𝒙𝑝 + 𝜕𝑦 𝒖˜ 𝒚𝑝 + 𝜕𝑧 𝒖˜ 𝒛𝑝 + 2𝜕𝑦 𝒖˜ 𝒙𝑝 𝜕𝑥 𝒖˜ 𝒚𝑝 + 2𝜕𝑥 𝒖˜ 𝒛𝑝 𝜕𝑧 𝒖˜ 𝒙𝑝 + 2𝜕𝑧 𝒖˜ 𝒚𝑝 𝜕𝑦 𝒖˜ 𝒛𝑝 ,(10)
where the superscript 𝑝 denotes that these values have been evaluated on 𝒙 𝒑 . Furthermore, for PTV data themeasured accelerations are available. This information can be used to enriched the pressure gradients in theoriginal measurement points 𝒙 𝑘 , these are taken into account by simply stacking them on top of the previous
pressure gradients:
𝜕𝑥 𝒑 = (𝜕𝑥 𝒑 𝑝 , −𝒂 𝒙 + 𝜈Δ𝒖˜ 𝒙 ) ;𝜕𝑦 𝒑 = 𝜕𝑦 𝒑 𝑝 , −𝒂 𝒚 + 𝜈Δ𝒖˜ 𝒚 ;𝜕𝑧 𝒑 = (𝜕𝑧 𝒑 𝑝 , −𝒂 𝒛 + 𝜈Δ𝒖˜ 𝒛 ) .

(11)
(12)
(13)
(14)

Finally, the cost function 5 can be rewritten as:
𝐽 (𝒘; 𝒙 𝑝 ) =∥𝜕𝑥 𝒑 − 𝜕𝑥 𝒑˜ ∥ 22
∥𝜕𝑥 𝒑∥ 22+∥𝜕𝑦 𝒑 − 𝜕𝑦 𝒑˜ ∥ 22
∥𝜕𝑦 𝒑∥ 22+∥𝜕𝑧 𝒑 − 𝜕𝑧 𝒑˜ ∥ 22
∥𝜕𝑧 𝒑∥ 22+∥Δ 𝒑 𝑝 − Δ 𝒑˜ 𝑝 ∥ 22
∥Δ 𝒑 𝑝 ∥ 22

.

(15)

Similar to the prior case the problem is linear and therefore efficiently handled with a conjugate gradientsolver. The resulting pressure is always relative lacking information about the absolute pressure. Therefore,a reference pressure point is required to evaluate the absolute pressures.

3

Test cases

To validate the method and prove its concept, two test cases are discussed. The first one 3.1 is purely synthetic
to allow a complete flow field resolution and test the methodology against the ground truth. Successively themethod is applied to an experimental investigation of a convective cell 3.1.

3.1

Synthetic Test Case

The test case has been selected for many reasons. Firstly, the simulation used to generate the synthetic
test case must preferably be a Direct Numerical Simulation (DNS) to have a high resolution. Secondly, areasonable Reynolds number must be achieved by the flow. The choice is to sample the JHTD Li et al. (2008)
and in particular, the channel flow Graham et al. (2016), this channel has the following sizes 8𝜋 × 2ℎ × 3𝜋[𝑚 3 ]with ℎ = 1[𝑚] . The first length refers to the stream-wise direction (𝑥 ), the second to the wall-normal direction( 𝑦 ) and the last to the span-wise direction (𝑧). The bulk velocity is 𝑈∞ = 1[𝑚/𝑠] and the Reynolds number
based on the wall-normal length is 𝑅𝑒 = 𝑈∞ 2ℎ/𝜈 = 4 · 104 . The synthetic measurement volume is placed at
19.75[𝑚] in the stream-wise direction with a size 0.2 × 0.2 × 0.2[𝑚 3 ] sharing the vertical plane of symmetry
with the channel. In this volume, approximately 20000 synthetic particles are randomly inserted and thenconvected with the flow. The positions are stored every 0.013[𝑠] = 50 · 𝑡 𝐷 𝑁 𝑆 , where 𝑡 𝐷 𝑁 𝑆 is the time step of
the DNS simulation. Assuming a setup with four 1024 × 1024 pixel cameras the number of particles per pixel( 𝑝 𝑝 𝑝 ) equals 𝐿 𝑥 × 𝐿 𝑦 × 𝐿 𝑧 = 0.02[particles/pixel3 ] . Synthetic noise with a standard deviation of 𝜎 = 0.5
pixel is added to nominal particle image positions.
Finally, the PTV velocities and accelerations are retrieved with a central finite difference scheme intime, starting from the noisy positions of the particles. These velocities have been reconstructed using threedifferent methods: linear interpolation, RBF (fit) and PI-RBF. The RBF and PI-RBFs used a regular grid of

𝑛𝑏 = 𝑛 𝑥 × 𝑛 𝑦 × 𝑛 𝑧 = 27 × 27 × 27 = 19683, in a volume slightly bigger than the measurement volume, every
dimension of the volume multiplied by 1.1 to ensure a smooth behaviour at the boundaries. The shape factor
is decided by a simple formula 𝑐 = 3 · 1.1 · 𝑚𝑎𝑥(𝐿 𝑥 /𝑛 𝑥 , 𝐿 𝑦 /𝑛 𝑦 , 𝐿 𝑧 /𝑛 𝑧 ) this shape factor ensures that for any
RBF in any axis direction, there is at least 6 other RBFs under its dome; the number of RBFs under eachother can be tweaked.
The second test case consists of a convective cell of approximately 60 × 60 × 60[𝑐𝑚 3 ] , in a plexiglass cubewith a heat source at the bottom and sealed at the top by a removable lid. The measurement setup includes
four FlowSense FCX-5M-124 cameras, with a maximum frame rate of 124𝐻𝑧 at full resolution. Higher
frame rates can be achieved by reducing the Region Of Interest (ROI) of the camera sensor. The desired
sampling frequency was 200𝐻𝑧 and to achieve this the ROI was reduced from 2048 × 2448 to 1906 × 1000
pixel. A total of 3000 frames was acquired, giving a run time of 15𝑠. The lenses used were 6.5𝑚𝑚 focal
length Kowa Optronics, and the resulting magnification was small enough to avoid the use of Scheimpflug
correction.
The convection inside the cell is driven by three resistors in series releasing 15𝑊 . The total measurement
volume is about 30 × 16 × 11[𝑐𝑚 3 ] in the top centre of the volume. HFSBs are released into the cell andilluminated by a 100𝑚𝐽 pulse from a 5 × 15 LED array and allowed to settle until the initial momentuum isdissipated. The resulting particle images have a high signal-to-noise ratio (1(a)). The tracks are computed byusing the Tomo-PTV routine of Dynamic Studio (DS) 8.0 and the raw tracks are smoothed by using a SavitzyGolay filter with a third-order polynomial on a window of 10 time steps; the resulting tracks are shown in
1(b). The final average particles tracked per snapshot are about 14000 resulting in an average particle densityof 0.01 particles per pixel(ppp). The RBFs are collocated on a regular grid of 𝑛𝑏 = 41 × 22 × 15 = 13530,

(a)

(b)

Figure 1: Post processed images on the left and raw tracks on the rightslightly lower than the average number of particles. The shape factor was determined in the same way as
described in 3.1. For this test case, only the velocity has been reconstructed due to the nature of the flow.
In a buoyancy-driven flow it is possible to invoke the Boussinesq approximation (see Tritton (1977)), whichassumes small changes in the density field. Therefore, to obtain the pressure the temperature should beknown in the whole domain, which goes beyond the scope of this contribution.

4

Results

4.1

Synthetic Test Case

The implementation was tested by comparing the different Probability Density Functions (PDFs) of theerror relative to the ground truth (DNS data), on a regular grid with 287496 (66 × 66 × 66) points at one
specific snapshot. In general, RBF methods outperform linear interpolation. The difference is evident if themain velocity component and its derivatives are taken into account (Fig.2). Among the PDFs proposed in
Fig.2, the major improvements concerning the derivatives of the PI-RBFs are expected in 𝜕𝑥 𝑢 𝑥 2(b), because
this quantity is contained explicitly in the divergence-free operator. Nonetheless, the enhancements of 𝑢
(Fig.2(a)) and 𝜕𝑦 𝑢 (Fig.2(d)) are still remarkable.

Linear
RBF-PI
RBF

0.150
0.125

Linear
RBF-PI
RBF

0.10
0.08

0.100
0.06

0.075

0.04

0.050

0.02

0.025
0.000
−0.02

−0.01

0.00
(ux − uxDN S ) /U∞

0.01

0.02

0.00

−0.04

(a) PDF 𝑢 𝑥

0.10

0.10
0.08

0.06

0.06

0.04

0.04

0.02

0.02
−0.04

−0.02
0.00
0.02
(∂y ux − ∂y uxDN S ) · hyDN S /U∞(c) PDF 𝜕𝑦 𝑢 𝑥

0.04

Linear
RBF-PIRBF fit

0.12

0.08

0.00

0.04

(b) PDF 𝜕𝑥 𝑢 𝑥Linear
RBF-PI
RBF

0.12

−0.02
0.00
0.02
(∂x ux − ∂x uxDN S ) · hxDN S /U∞

0.00

−0.04

−0.02
0.00
0.02
∂z ux − ∂z uDN S · hzDN S /U∞

0.04(d) PDF 𝜕𝑧 𝑢 𝑥Figure 2: PDF for the various quantity, where ℎ 𝑥𝐷 𝑁 𝑆 and ℎ 𝑦𝐷 𝑁 𝑆 are the grid step of the DNS in the x andy direction, at the fifth snapshot.

Statistics performs better for RBF-based methods in general as it is possible to deduce from Fig.3, where
the non-dimensional Reynolds stress 𝑢 ′𝑥 𝑢 ′+𝑥 is shown. The RBFs methods preserve all the turbulent energy
through the boundary layer while the linear interpolation loses a large amount of it.
10

DNS
RBF-PI
RBF
Linearux 0 ux 0+ [−]

8

6

4

2

0
0

25

50

75

100

125

150

175

200

y + [−]

Figure 3: Non-dimensional Reynolds stress 𝑢 ′𝑥 𝑢 ′+𝑥 .
The velocity 𝑢 𝑧 contour in the central spanwise plane is shown in Fig.4, for the DNS (Fig.4(a)), PIRBFs (Fig.4(c)) and linear interpolation (Fig.4(e)). This figure shows the capability of the PI-RBF, which
reconstructs smoother velocity than the linear interpolation.
PI-RBF performs significantly better than the linear interpolation, when derivatives are taken into
account as the vorticity magnitude ( ∥𝝎∥ 2 ) presented in Fig.4. Moreover, in Fig.4(f), some important errors
are revealed in the boundaries. Instead, the PI-RBFs do not show this problem.
Finally the pressure is shown in Fig.5, the top left shows a snapshot of the DNS pressure Fig,5(a), thetop right shows the corresponding PI-RBF pressure Fig,5(b) and the bottom shows the error Fig,5(c), The
topmost pictures are qualitatively similar. Nevertheless, an important source of error can be found on theboundaries, except on the lower boundary, representing the wall. This decrease in error in the lower boundaryis probably due to the reference pressure point, which is located exactly on the lower wall in the middle ofthe domain (i.e. [0.1, 0, 0.1] ).

0.200

0.200
0.16

0.150

0.175

0.00

0.125
y[m]

uz [m/s]

0.075

20

0.075
−0.08

0.050
0.025
0.000
0.00

30

0.100

0.050

0.05

0.10
x[m]

0.15

10

0.025

−0.16

0.000
0.00

0.20

0.05

(a) DNS

0.20

0.200
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0.175
0.150

40

0.00

0.125
y[m]

uz [m/s]

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50

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30

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20

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x[m]

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0.05

0

0.175

0.125
y[m]

0.15

(b) DNS

0.200

0.000
0.00

0.10
x[m]

kωk2 [s−1 ]

y[m]

0.125
0.100

40

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0.08

kωk2 [s−1 ]

0.175

50

0.000
0.00

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0.05

(c) PI-RBF

0.10
x[m]

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0.20

0

(d) PI-RBF

0.200

0.200
0.16

0.175

50

0.175
40

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y[m]

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uz [m/s]

y[m]

0.125

30

0.100
20

0.075
−0.08

0.050

kωk2 [s−1 ]

0.08

0.050
10

0.025
0.000
0.00

−0.16
0.05

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x[m]

(e) Linear

0.20

0.025
0.000
0.00

0.05

0.10

0.15

0.20

0

x[m]

(f) Linear

Figure 4: Contours of the instantaneous z-direction velocity component 𝑢 𝑧 (left) and the vorticity magnitude∥𝝎∥ 2 (right) for DNS (top), PI-RBF (middle) and Linear interpolation (bottom).

0.200

0.006

0.200

0.175

0.006

0.175
0.001

0.001
0.150

−0.009

0.075

y[m]

0.125

p[m2 /s2 ]−0.0040.100−0.0090.050−0.014

0.025

0.05

0.10

0.15

−0.0190.20−0.014

0.025
0.000
0.00

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x[m]

x[m]

(a) DNS(b) PI-RBF

0.200

0.005

0.175
0.004
0.150
0.125

0.003

0.100
0.002

0.075
0.050

|pDN S − pRBF |[m2 /s2 ]

0.000
0.00

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0.050

y[m]

y[m]

0.125

0.001
0.025
0.000
0.00

0.05

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0.000

x[m]

(c) Error

Figure 5: The instantaneous pressure

0.20−0.019p[m2 /s2 ]

0.150

4.2

Convective Cell

The experimental results of the convection show the typical structures of the rising plume that spreads
out when reaching the lid, with large re-circulation zones established around it. These structures can be
distinctively identify in Fig.6(a), here highlighted the average flow field and the Turbulent Kinetic Energy(TKE) Fig.6(b) of the flow derived from the PI-RBFs.
0.192

0.00384

0.168

0.003360.072−0.15 −0.10 −0.05 0.00
x[m]

0.05

0.10

0.15

0.048

0.00240
y[m]

0.096
0.00

kuk2 [m/s]

y[m]

0.120
0.05

0.002880.10
0.05

0.001920.00

0.00144
−0.1

0.024

0.0
x[m]

0.1

0.000

(a) Middle in-plane average velocities(c) PI-RBFs instantaneous Q-criterion (100velocity magnitude color-codedk[m2 /s2 ]

0.144

0.10

0.000960.000480.00000(b) TKE𝐻𝑧 2 ) (d) Linear instantaneous Q-criterion (100locity magnitude color-coded

𝐻𝑧 2 ) ve-

Figure 6: Convective flow results
Similar to the vorticity results for the synthetic test case, the Q-criterion derived from the linear interpolations is more noisy and the structures in the flow fields are less visible than in the PI-RBFs. These results
highlight the capabilities of using PI-RBFs in experiments see Fig.6(c) and Fig.6(d).

5Conclusions and outlooks

The PI-RBFs are applied on a synthetic turbulent channel flow data with added noise, this can be seen asa proof of concept. Comparisons with linear interpolation showed the velocities and especially all velocity
derivatives show a significant improvement in the accuracy.
Finally, the PI-RBFs method is applied to experimental data in a convection cell of about 0.2𝑚 3 using
neutrally buoyant HFSB, where it shows an improved capability of detecting structures and flow features. In
this case not only do the structures appear more smooth, but also seemingly more structures are detected.
This could be caused by the improved robustness of the PI-RBF in correcting errors due to the whole chain

of measurement and not only white noise. Therefore, work can be done to test the method not only with asynthetic test case with added white noise but also by preparing synthetic images to then post-process andretrieve the velocities from.
Future work will focus on non-linear solvers that can take into account the whole Navier-stokes equationto achieve proper resolution and improved results.

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