# A RANS approach to the Meshless Computation of Pressure Fields From
Image Velocimetry

Pietro Sperotto1∗ Sandra Pieraccini2 Miguel A. Mendez1 1von Karman Institute
for Fluid Dynamics, Sint-Genesius-Rode, Belgium
2Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino,
Italy
*Corresponding author<EMAIL_ADDRESS>
###### Abstract

We propose a 3D meshless method to compute mean pressure fields in turbulent
flows from image velocimetry. The method is an extension of the constrained
Radial Basis Function (RBF) formulation by Sperotto . (2022) to a Reynolds
Averaged Navier Stokes (RANS) framework. This is designed to handle both
scattered data as in Particle Tracking Velocimetry (PTV) and data in uniform
grids as in correlation-based Particle Image Velocimetry (PIV). The RANS
extension includes the Reynolds stresses into the constrained least square
problem. We test the approach on a numerical database featuring a Backward
Facing Step (BFS) with a Reynolds number of 6400 (defined with respect to the
inlet velocity and step height), obtained via Direct Numerical Simulation
(DNS).

20th LISBON Laser Symposium 2022

Keywords: Pressure from PIV and PTV, Radial Basis Functions, Meshless
integration of PDEs.

##  1 Introduction

Many methods have been proposed to compute pressure fields from image
velocimetry in the last decade. These can be broadly classified into Eulerian
and Lagrangian methods depending on how the material acceleration is computed.
Eulerian methods are based on the available velocity fields, which can be
instantaneous or time-averaged. These can be further classified into
directional method, integrating Navier Stokes equation (eg. Jakobsen . (1997);
Köngeter (1999); Liu Katz (2006); Wang . (2019)), and global method, solving
the pressure Poisson equation (eg. Gurka . (1999); Ghaemi . (2012); Pan .
(2016)). Lagrangian methods are based on the computation of particle
trajectories and have been primarily fostered by recent advances in 3D
tracking techniques (Schanz . (2016)). These can be further distinguished into
techniques that interpolate the particle acceleration onto a Cartesian grid
(Gesemann ., 2016; Huhn ., 2016, 2018; Schneiders Scarano, 2016) and
techniques that integrate the pressure on scattered data such as the Voronoi
integration proposed by Neeteson Rival (2015).

All of the aforementioned methods rely on ‘classic‘ mesh-based numerical
methods (e.g. Finite Differences or Finite Elements) to solve the Partial
Differential Equations (PDE) involved, namely the Navier Stokes Equation or
the Poisson equation. Recently, we proposed a meshless approach based on
Radial Basis Functions (Sperotto ., 2022) for integrating the pressure field
from scattered velocity measurements in an incompressible and laminar flow.
This approach builds on the vast literature of RBF based PDE solvers (Kansa,
1990; Fornberg Flyer, 2015; Šarler, 2005; Chen Tanaka, 2002; Chen, 2003;
Šarler, 2007) and consists in solving two linear least square problems with
differential constraints. The first is the regression of the velocity field
constrained by the continuity equation and the relevant boundary conditions;
the second is the integration of the pressure Poisson equation given the
regressed velocity fields.

In this work, we extend the previous contribution in Sperotto . (2022) to the
Reynolds Averaged Navier Stokes (RANS) formulation firstly proposed by Gurka .
(1999). The pressure integration consists in solving the pressure Poisson
equation in the RANS problem, i.e. including the contribution of the Reynolds
stresses. Assuming that the available dataset is statistically sufficiently
resolved to compute the Reynolds stresses, these are regressed using an RBF
expansion as done for the velocity fields. Both the RBF approximations of the
Reynolds stresses and velocity fields are then used in the pressure
computation.

We introduce the RANS based pressure computation problem and the RBF-based
integration approach in Section 2. We test our approach on the flow past a
Backward Facing Step (BFS) at a moderately large Reynolds number. The dataset
was constructed by sampling a Direct Numerical Simulation (DNS), thus
providing high-fidelity and statistically resolved data on the mean velocity
and mean pressure fields. This allows testing the accuracy of the RBF
regression and pressure computation. The test case is described in Section 3.
Finally, section 4 reports on the comparison between the RBF regression and
pressure integration and the ground truth, while section 5 closes with
conclusions and outlooks.

##  2 Methodology

The starting point of the method is the RANS formulation, which we briefly
recall in section 2.1. Three linear least square problems are set. The first
two are (1) the RBF regression of the (mean) velocity field and (2) the RBF
regression of the Reynolds stresses. Both are carried out with differential
constraints. The third problem consists in (3) solving the pressure Poisson
equation using the (analytic) results of the previous two steps.

While the solution of problem (1) and (3) are extensively described in
Sperotto . (2022), the step (2) constitute a major novelty and is briefly
illustrated in section 2.3. The fundamentals of the RBF-regression and
meshless Poisson solver are briefly recalled in 2.2.

###  2.1 The RANS formulation

The RANS equation are obtained by introducing the Reynolds decomposition
$\bm{u}=\overline{\bm{u}}(\bm{x})+\bm{u}^{\prime}(\bm{x},t)$, where
$\overline{\bm{u}}(\mathbf{x})=(\overline{u},\overline{v},\overline{w})$ is
the averaged velocity field and
$\bm{u}^{\prime}(\bm{x},t)=(u^{\prime},v^{\prime},w^{\prime})$ is the
fluctuating (time dependent) field. For the incompressible flow of a Newtonian
fluids with constant properties, introducing this decomposition in the Navier
stokes equation and averaging yields the RANS equations:

$\rho\left(\overline{\bm{u}}\cdot\nabla\right)\overline{\bm{u}}=-\nabla\overline{p}+\mu\Delta\overline{\bm{u}}-\rho\nabla\cdot\mathbf{R}\left(\bm{u}^{\prime},\bm{u}^{\prime}\right),$
(1)

where $p$ the pressure field, $\rho$ is the density, $\mu$ the dynamic
viscosity and
$\mathbf{R}\left(\bm{u}^{\prime},\bm{u}^{\prime}\right)=\overline{u^{\prime}_{i}u^{\prime}_{j}}$
is the Reynolds stresses tensor, with the indices $i,j=1,2,3$ referring to the
three components. This collects the correlation between the fluctuating
components of the velocity field and introduces additional stresses used to
model the mean flow’s non-linear diffusion due to turbulence. Taking the
divergence on both sides and using the continuity equation for the mean flow
(i.e. $\nabla\cdot\overline{\bm{u}}=0$) yields the pressure Poisson equation
in the RANS context. Together with its boundary conditions, this reads Pope
(2000)

$\begin{cases}\Delta\overline{p}=-\rho\left(\nabla\cdot\left(\overline{\bm{u}}\cdot\nabla\overline{\bm{u}}\right)+\nabla\cdot\left(\nabla\cdot\mathbf{R}\left(\bm{u}^{\prime},\bm{u}^{\prime}\right)\right)\right)\quad\bm{x}\in\Omega\\\
\partial_{\bm{n}}\overline{p}=\nabla\overline{p}\cdot\bm{n}=\rho\left(-\left(\overline{\bm{u}}\cdot\nabla\right)\overline{\bm{u}}-\nabla\cdot\mathbf{R}\left(\bm{u}^{\prime},\bm{u}^{\prime}\right)+\nu\Delta\overline{\bm{u}}\right)\cdot\bm{n}\quad\bm{x}\in\Gamma_{N}\\\
\overline{p}=\overline{p_{D}}\quad\bm{x}\in\Gamma_{D}.\end{cases}$ (2)

where $\Omega\subset\mathbb{R}^{3}$ is the integration domain with boundaries
$\partial\Omega$. The Dirichlet and Neumann condition are applied respectively
in $\Gamma_{D}$ and $\Gamma_{N}$, which are such that
$\Gamma_{D}\cup\Gamma_{N}=\partial\Omega$.

In the context of pressure integration from measured velocity fields, the
Dirichlet conditions are usually prescribed by a set of pressure probes (e.g.
pressure taps or Pitot tubes) while the Neumann boundary conditions are
evaluated by projecting the RANS equations into the relevant boundaries. The
reader is referred to Pan . (2016); Faiella . (2021) for an extensive
discussion on the impact of boundary conditions in the pressure integration
from image velocimetry.

###  2.2 RBF Regression and Meshless PDE Integration

The RBF regression consists in finding an approximation of a function as a
linear combination of RBFs. That is, given a function
${f}(\bm{x}):\mathbb{R}^{3}\rightarrow\mathbb{R}$, we seek the set of weights
$w_{i}$ such that

${f}(\bm{x})\simeq\tilde{f}(\bm{x})=\sum_{i=1}^{n_{c}}w_{i}\varphi\left(\bm{x}\mid\bm{x_{i}}^{*},c_{i}\right)=\sum_{i=1}^{n_{c}}w_{i}\varphi_{i}$
(3)

where $c_{i}$ and $\bm{x_{i}}^{*}$ are the shape factor and the collocation
point of the i-th basis functions $\varphi_{i}$ and $n_{c}$ is the number of
bases involved. As in Sperotto . (2022), we here consider isotropic Gaussians
as basis functions.

Assuming that values of $f(\bm{x})$ are given in a set of $n_{p}$ points such
that $\bm{f}=f\left(\bm{x}_{j}\right)=f_{j}$ with $j=1,\dots,n_{p}$, the
regression consists in finding the weights $w_{i}$ that minimize the least
square error between the approximation and the available points. Arranging the
basis functions into a matrix
$\mathbf{\Phi}\left(\bm{X}\right)\in\mathbb{R}^{n_{p}\times n_{c}}$, with
$\bm{X}\in\mathbb{R}^{3\times n_{p}}$ the coordinates of the points
$\mathbf{x}_{j}$ in which data is available, equation (3) can be written as
$\bm{f}\approx\mathbf{\Phi}\left(\bm{X}\right)\cdot\bm{w}$ where $\bm{w}$ is
the vector collecting the $n_{c}$ weights $w_{i}$.

Denoting as $\mathcal{L}\\{f\\}(\mathbf{x})$ a differential operator acting on
$f$, one has
$\mathcal{L}\\{f\\}(\mathbf{x})=\mathcal{L}\\{\mathbf{\Phi}\\}\left(\bm{x}\right)\cdot\bm{w}$.
A differential constraint is thus a linear constraint on $\bm{w}$, here
denoted $\mathcal{L}\\{\mathbf{\Phi}\\}\cdot\bm{w}=\bm{f}_{L}$. Similarly,
Dirichlet boundary conditions on $\mathbf{X}_{D}\in\Gamma_{D}$ can be written
as $\mathbf{\Phi}\left(\mathbf{X}_{D}\right)\cdot\bm{w}=\bm{f}_{D}$ and
Neumann boundary conditions on $\mathbf{X}_{N}\in\Gamma_{N}$ can be written as
$\partial_{n}\mathbf{\Phi}\left(\mathbf{X}_{N}\right)\cdot\bm{w}=\bm{f}_{N}$,
where $\partial_{n}$ denotes derivatives normal to the boundaries.

The RBF regression with differential constraints with Dirichlet and Neumann
boundary conditions is thus a quadratic programming problem with linear
constraints. Using Lagrange multipliers, the solution consists in finding the
weights $\bm{w}$ and the multipliers
$\bm{\lambda}=(\bm{\lambda}_{\mathcal{L}},\bm{\lambda}_{D},\bm{\lambda}_{N})$
that minimize the augmented cost function

$J^{*}(\bm{w},\bm{\lambda})=||\bm{f}-\bm{\Phi}(\bm{X})\cdot\bm{w}||^{2}_{2}+\bm{\lambda}^{T}_{\mathcal{L}}\biggl{(}\mathcal{L}\\{\mathbf{\Phi}\\}\cdot\bm{w}-\bm{f}_{L}\biggr{)}+\bm{\lambda}^{T}_{\mathcal{D}}\biggl{(}\mathbf{\Phi}\left(\Gamma_{D}\right)\cdot\bm{w}-\bm{f}_{D}\biggr{)}+\bm{\lambda}^{T}_{\mathcal{N}}\biggl{(}\mathbf{\Phi}\left(\Gamma_{N}\right)\cdot\bm{w}-\bm{f}_{N}\biggr{)}$
(4)

where $||\bullet||_{2}$ denotes the $l_{2}$ norm.

In the regression of the velocity field, differential constraints are used to
impose the divergence-free conditions; the Dirichlet conditions are used to
set no-slip on walls and Neumann conditions are used on ’outlet’ patches. In
the solution of the pressure Poisson equation, the basis functions are the
Laplacian of the RBFs, and the constraints are used to set the boundary
conditions for the pressure integration. The reader is referred to Sperotto .
(2022) for more details on the least square problem formulation for scalar and
vector fields and the numerical methods implemented for their solution.

###  2.3 The Regression of the Reynolds stresses

The problem of regressing the Reynolds stresses is analogous to the problem of
regressing the velocity field. In the formulation implemented in this work, no
simplifications are considered for the Reynolds stresses. Therefore, this
regression consists in regressing six functions, that is the entries in the
tensor
$\mathbf{R}\left(\bm{u}^{\prime},\bm{u}^{\prime}\right)=\overline{u^{\prime}_{i}u^{\prime}_{j}}$.
The problem can be considerably simplified if turbulence is assumed to be
isotropic (then $\mathbf{R}$ is diagonal) or isotropic _and_ homogeneous (then
$\mathbf{R}$ is a multiple the identity matrix).

The regression of these terms were constrained with the Dirichlet condition
$\overline{u^{\prime}_{i}u^{\prime}_{j}}=0\,\forall i,j=1,2,3$ at walls.
Moreover, since the velocity fluctuations must vanish sufficiently close to
the walls, it is also possible to show that all derivatives of the Reynolds
stresses must also vanish at walls, that is
$\partial_{{x}_{k}}\overline{u^{\prime}_{i}u^{\prime}_{j}}=0\,\forall
i,j,k=1,2,3$.

##  3 Select Test Case

We create a synthetic 3D PTV database for the flow past a Backward Facing Step
(BFS). The data was obtained via Direct Numerical Simulation (DNS) by Oder .
(2019). This simulation analyzes the heat transfer in a low Prandtl number
fluid ($Pr=0.005$) and Reynolds number, based on the inlet velocity $U_{0}$
and step height $h$, of $Re=\rho hU_{0}/\mu=6400$. The main geometrical
parameters are shown in Figure 1.

The dataset was provided in dimensionless form, with velocities normalized
with respect to the inlet velocity and spatial coordinates normalized with
respect to the step height. Therefore, we take the freedom of setting the
inlet velocity to $U_{0}=1$m/s, and the step height to $h=2$cm. Taking
$\rho=1$ kg/m3, we then consider $\mu/\rho=3.125\cdot 10^{-5}$m2/s to keep the
same Reynolds number. The remaining relevant dimensions, as shown in Figure 1,
respect the proportions of the computational domain by Oder . (2019). The
domain extends $3.6$ cm in the $z$ direction. Inlet, outlet and wall patches
are shown with violet, blue and green lines respectively.

34 cm14 cm2 cm12 cm3.6 cmStepxyzWalls in greenOutlet in blueInlet in
violetPressure Taps

Figure 1: Domain of the analyzed configuration, namely the flow past a BFS via
DNS by Oder . (2019). The region of interest simulating an experimental
acquisition is shown in red. The black markers illustrate the position of $10$
pressure taps, installed with a spacing of $1.4$cm, used to aid the pressure
computation from the velocity fields.

We construct the 3D PTV dataset assuming that the region of interest for the
measurement is the one in red in figure 1. The measurement volume is $8$mm
thick in the z-direction and is located in the centre of the channel along z.
To aid the pressure computation from the image velocimetry, we assume that
$10$ pressure taps are installed on the bottom wall (black markers in figure
1) with a spacing of $1.4$cm.

This work considers only averaged quantities, hence mean velocity, mean
pressure, and Reynolds’s stress fields. These were randomly sampled on $8400$
points from the available computational grid, i.e. avoiding any form of
interpolation in the data generation. Considering images with 1024 x 2048
pixels, this leads to a particle density of $3.4\cdot 10^{-3}$ ppp. Figure 2
shows the mean velocity field of the obtained PTV dataset. This scattered
sampling of the averaged fields can be pictured as the result of an ensemble
PTV averaging as proposed by Agüera . (2016). Our meshless method works
equally well on a uniform grid (such as obtained via PIV), but we stick to the
scattered sampling approach to test the accuracy of the regression without the
interfering effect of interpolation in the data preparation. Similarly, we
here take the Reynolds stresses in these points from the DNS to evaluate the
regression performances on the ground truth data. However, to make the data
more realistic, we add a random white noise of $5$% on the velocity, pressure
and Reynolds stresses fields as well as on the sampling points simulating
pressure taps.

Figure 2: Plot of the sampled mean flow field in the middle plane. All the
available points are shown; this allows visualizing the (modest) seeding
density used in the investigated test case.

##  4 Results

The results are given in terms of the euclidean norm
$E=\|f\left(\mathbf{X}\right)-\tilde{f}\left(\mathbf{X}\right)\|_{2}$ and the
relative euclidean norm
$\varepsilon=\|f\left(\mathbf{X}\right)-\tilde{f}\left(\mathbf{X}\right)\|_{2}/\|f\left(\mathbf{X}\right)\|_{2}$
in Table 1. These result show that some variables
($\overline{w},\overline{u^{\prime}w^{\prime}}$ and
$\overline{v^{\prime}w^{\prime}}$) have large values of $\varepsilon$.
However, these fields still an acceptable $E$, hence the main discrepancy is
due to the low value of these quantities for the investigated test case.

The result of the regression for the mean velocity are shown in Figure 3
(right) and compared with the DNS data (left). These are indistinguishable.
Figure 4 shows the same comparison for the velocity component $\overline{w}$,
which particularly low values because of the bi-dimensionality of the flow.
The reconstruction appears slightly smoothed but captures the main features.
Finally, figures 5 and 6 shows the same comparison for the turbulent kinetic
energy and pressure fields. Considering the moderate seeding density and the
realistic noise levels, these results are considered largely satisfactory.

| $u$ | $v$ | $w$ | $\overline{u^{\prime}u^{\prime}}$ | $\overline{u^{\prime}v^{\prime}}$ | $\overline{u^{\prime}w^{\prime}}$ | $\overline{v^{\prime}v^{\prime}}$ | $\overline{v^{\prime}w^{\prime}}$ | $\overline{w^{\prime}w^{\prime}}$ | $p$
---|---|---|---|---|---|---|---|---|---|---
$\varepsilon$ | 0.026 | 0.054 | 0.144 | 0.036 | 0.019 | 0.314 | 0.019 | 0.172 | 0.063 | 0.038
$E$ | 0.571 | 0.156 | 0.206 | 0.111 | 0.021 | 0.013 | 0.033 | 0.028 | 0.128 | 0.668

Table 1: Table of errors.

The resulting in-plane velocities are shown in Fig. 3. Instead, the pressure
reconstruction is displayed in Fig. 6. The results are in agreement with the
DNS flow field. Even though, they are compared over a uniform grid of size
$100\times 100$ in one plane resulting in $10000$ points. This value exceeds
the original PTV data, sampled over the whole volume, by almost $2000$ points.

Figure 3: In plane velocity components $(u,v)$. DNS (left) and reconstructed
(right)

Figure 4: Span-wise velocity component $w$. DNS (left) and reconstructed
(right)

Figure 5: Turbulent kinetic energy $k$. DNS (left) and reconstructed (right)

Figure 6: Pressure $p$. DNS (left) and reconstructed (right)

##  5 Conclusion and future perspectives

We proposed the first RANS extension of the RBF meshless integration framework
for computing pressure fields from image velocimetry. This allows extending
the meshless pressure computation to the mean pressure fields in turbulent
flows. The method requires the statistical convergence of the velocimetry
measurements and extends the work presented in Sperotto . (2022) by including
the (constrained) regression of the Reynolds stresses.

The meshless RANS integration was tested on a synthetic dataset constructed
from DNS data, and no simplifying assumptions on the Reynolds stresses were
considered (i.e. the turbulence is non-homogeneous and non-isotropic). The
results of the pressure computation were excellent. Ongoing activities are
focused on applying the method to PIV experimental data.

## References

  * Agüera . (2016) aguera2016ensembleAgüera, N., Cafiero, G., Astarita, T. Discetti, S.  2016\.  Ensemble 3D PTV for high resolution turbulent statistics Ensemble 3d ptv for high resolution turbulent statistics.  Measurement Science and Technology2712124011.
  * Chen (2003) Chen2003Chen, W.  2003\.  New RBF Collocation Methods and Kernel RBF with Applications New RBF collocation methods and kernel RBF with applications.  Lecture Notes in Computational Science and Engineering Lecture notes in computational science and engineering ( 75–86).  Springer Berlin Heidelberg.  10.1007/978-3-642-56103-0_6
  * Chen Tanaka (2002) Chen2002Chen, W. Tanaka, M.  2002feb.  A meshless, integration-free, and boundary-only RBF technique A meshless, integration-free, and boundary-only RBF technique.  Computers & Mathematics with Applications433-5379–391.  10.1016/s0898-1221(01)00293-0
  * Faiella . (2021) Faiella2021Faiella, M., Macmillan, CGJ., Whitehead, JP. Pan, Z.  2021may.  Error propagation dynamics of velocimetry-based pressure field calculations (2): on the error profile Error propagation dynamics of velocimetry-based pressure field calculations (2): on the error profile.  Measurement Science and Technology328084005.  10.1088/1361-6501/abf30d
  * Fornberg Flyer (2015) Fornberg2015Fornberg, B. Flyer, N.  2015apr.  Solving PDEs with radial basis functions Solving PDEs with radial basis functions.  Acta Numerica24215–258.  10.1017/s0962492914000130
  * Gesemann . (2016) GesemanGesemann, S., Hunh, F., Schanz, D. Schröder, A.  2016\.  From noisy particle tracks to velocity, acceleration and pressure fields using B-splines and penalties From noisy particle tracks to velocity, acceleration and pressure fields using b-splines and penalties.  Int. Symp. on Applications of Laser Techniques to Fluid Mechanics. Int. symp. on applications of laser techniques to fluid mechanics.
  * Ghaemi . (2012) Ghaemi2012Ghaemi, S., Ragni, D. Scarano, F.  2012oct.  PIV-based pressure fluctuations in the turbulent boundary layer PIV-based pressure fluctuations in the turbulent boundary layer.  Experiments in Fluids5361823–1840.  10.1007/s00348-012-1391-4
  * Gurka . (1999) gurka1999computationGurka, R., Liberzon, A., Hefetz, D., Rubinstein, D. Shavit, U.  1999\.  Computation of pressure distribution using PIV velocity data Computation of pressure distribution using piv velocity data.  Workshop on particle image velocimetry Workshop on particle image velocimetry ( 2, 1–6).
  * Huhn . (2016) Huhn2016Huhn, F., Schanz, D., Gesemann, S. Schröder, A.  2016sep.  FFT integration of instantaneous 3D pressure gradient fields measured by Lagrangian particle tracking in turbulent flows FFT integration of instantaneous 3d pressure gradient fields measured by lagrangian particle tracking in turbulent flows.  Experiments in Fluids579.  10.1007/s00348-016-2236-3
  * Huhn . (2018) Huhn2018Huhn, F., Schanz, D., Manovski, P., Gesemann, S. Schröder, A.  2018apr.  Time-resolved large-scale volumetric pressure fields of an impinging jet from dense Lagrangian particle tracking Time-resolved large-scale volumetric pressure fields of an impinging jet from dense lagrangian particle tracking.  Experiments in Fluids595.  10.1007/s00348-018-2533-0
  * Jakobsen . (1997) Jakobsen1997Jakobsen, ML., Dewhirst, TP. Greated, CA.  1997dec.  Particle image velocimetry for predictions of acceleration fields and force within fluid flows Particle image velocimetry for predictions of acceleration fields and force within fluid flows.  Measurement Science and Technology8121502–1516.  10.1088/0957-0233/8/12/013
  * Kansa (1990) Kansa1990Kansa, E.  1990\.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—i surface approximations and partial derivative estimates.  Computers & Mathematics with Applications198-9127–145.  10.1016/0898-1221(90)90270-t
  * Köngeter (1999) Kngeter1999PIVWHKöngeter, J.  1999\.  PIV with high temporal resolution for the determination of local pressure reductions from coherent turbulence phenomena Piv with high temporal resolution for the determination of local pressure reductions from coherent turbulence phenomena.  Experiments in Fluids29.
  * Liu Katz (2006) Liu2006Liu, X. Katz, J.  2006may.  Instantaneous pressure and material acceleration measurements using a four-exposure PIV system Instantaneous pressure and material acceleration measurements using a four-exposure PIV system.  Experiments in Fluids412227–240.  10.1007/s00348-006-0152-7
  * Neeteson Rival (2015) Neeteson2015Neeteson, NJ. Rival, DE.  2015feb.  Pressure-field extraction on unstructured flow data using a Voronoi tessellation-based networking algorithm: a proof-of-principle study Pressure-field extraction on unstructured flow data using a voronoi tessellation-based networking algorithm: a proof-of-principle study.  Experiments in Fluids562.  10.1007/s00348-015-1911-0
  * Oder . (2019) ODER2019118436Oder, J., Shams, A., Cizelj, L. Tiselj, I.  2019\.  Direct numerical simulation of low-Prandtl fluid flow over a confined backward facing step Direct numerical simulation of low-prandtl fluid flow over a confined backward facing step.  International Journal of Heat and Mass Transfer142118436.  https://doi.org/10.1016/j.ijheatmasstransfer.2019.118436
  * Pan . (2016) Pan2016Pan, Z., Whitehead, J., Thomson, S. Truscott, T.  2016jul.  Error propagation dynamics of PIV-based pressure field calculations: How well does the pressure Poisson solver perform inherently? Error propagation dynamics of PIV-based pressure field calculations: How well does the pressure poisson solver perform inherently?  Measurement Science and Technology278084012.  10.1088/0957-0233/27/8/084012
  * Pope (2000) Pope2000Pope, SB.  2000\.  Turbulent Flows Turbulent flows.  Cambridge University Pr.
  * Šarler (2007) SarlerŠarler, B.  2007\.  From Global to Local Radial Basis Function Collocation Method for Transport Phenomena From global to local radial basis function collocation method for transport phenomena.  Advances in Meshfree Techniques Advances in meshfree techniques ( 257–282).  Springer Netherlands.  10.1007/978-1-4020-6095-3_14
  * Schanz . (2016) Schanz2016Schanz, D., Gesemann, S. Schröder, A.  2016apr.  Shake-The-Box: Lagrangian particle tracking at high particle image densities Shake-the-box: Lagrangian particle tracking at high particle image densities.  Experiments in Fluids575.  10.1007/s00348-016-2157-1
  * Schneiders Scarano (2016) Schneiders2016Schneiders, JFG. Scarano, F.  2016aug.  Dense velocity reconstruction from tomographic PTV with material derivatives Dense velocity reconstruction from tomographic PTV with material derivatives.  Experiments in Fluids579.  10.1007/s00348-016-2225-6
  * Sperotto . (2022) Sperotto2022Sperotto, P., Pieraccini, S. Mendez, MA.  2022\.  A Meshless Method to Compute Pressure Fields from Image Velocimetry A meshless method to compute pressure fields from image velocimetry.  Measurement Science and Technology.  10.1088/1361-6501/ac70a9
  * Wang . (2019) Wang2019Wang, J., Zhang, C. Katz, J.  2019mar.  GPU-based, parallel-line, omni-directional integration of measured pressure gradient field to obtain the 3D pressure distribution GPU-based, parallel-line, omni-directional integration of measured pressure gradient field to obtain the 3d pressure distribution.  Experiments in Fluids604.  10.1007/s00348-019-2700-y
  * Šarler (2005) cmesŠarler, B.  2005\.  A Radial Basis Function Collocation Approach in Computational Fluid Dynamics A radial basis function collocation approach in computational fluid dynamics.  Computer Modeling in Engineering & Sciences72185–194.  10.3970/cmes.2005.007.185