Assessment of spatial derivatives determined from scattered 3D PTV data
er, S. Blaser
Abstract Based on a linear regression model, we propose
a numerical method to determine the spatial derivatives
from scattered 3D PTV (particle-tracking velocimetry)
data.Various quantities allowing an assessment of the
numerically calculated gradients are introduced and their
reliability is investigated.The performance of the numer-
ical scheme and of the ``quality estimators'' was examined
for different synthetic data sets obtained from Burgers'
vortex.The energy dissipation was computed from ex-
perimental PTV measurements performed for a forward-
facing step con®guration.
With the progress made in the development of experi-
mental facilities, such as CCD cameras and computers, it
has become feasible to accurately measure the motion of
particles.Particle-tracking velocimetry (PTV) is such a
measurement technique, which yields the particle coordi-
nates and velocities in a three-dimensional ¯ow volume V
over consecutive time steps (Maas et al.1993; Malik et al.
1993; Virant and Dracos 1997).The particles are, however,
not uniformly dispersed in V, which impedes, for example,
the determination of spatial velocity gradients.Neverthe-
less, many quantities which are of physical interest, such
as vorticity, shear rate, and energy dissipation, depend on
the velocity derivatives.In the present study, we are par-
ticularly interested in spatial gradients which occur along
trajectories of single particles.
One approach of determining gradients is to ®rst
interpolate the scattered data onto a regular grid and then
to compute the derivatives using a standard scheme (e.g.,
nez 1987, Malik and Dracos 1995).However,
this procedure involves an additional interpolation error
which is superimposed on the measured velocities and
which may even be increased when the vector ®eld is
differentiated.In addition, in order to resolve appropri-
ately the length scales, the particle density tends to be
high, which makes particle tracking in the three-dimen-
sional space rather dif®cult.
If the velocity vectors are randomly located in a plane,
Cowen and Monismith (1997) suggest the following
method to determine the dissipation e in 2D: the gradients,
e.g., along the x
-direction at (x
), are calculated by
taking the velocity vector at this point and a velocity
vector which is found within the square of side length
0.05 á g and center (x
), where g is the Kolmogorov
length scale and c ³ 1 a constant.This procedure has to be
done iteratively, as g depends on e through the equation
, where m is the dynamic viscosity.This
method may be appropriate for measurements in a plane,
in which case a suf®cient particle density can be readily
guaranteed.Again, 3D PTV works reliably only at mod-
erate particle concentrations.Our current PTV system can
track approximately 1000 vectors in a minimum volume of
; thus the spatial resolution is, according to the
Nyquist sampling criterion, 2 mm.
In the following, we will propose a numerical procedure
to determine all nine components of the velocity gradient
tensor ,u from scattered 3D PTV data.The main idea of
the method is to expand the velocity ®eld into a Taylor
series up to ®rst order and to use a linear regression model
for calculation of ,u.
It is clear that any determination of a physical quantity
should be accompanied by a proper error analysis.For this
purpose, we introduce several estimators which allow an
assessment of the ``quality'' of the numerically computed
gradients.Different synthetic data sets based on Burgers'
vortex were produced in order to examine the perfor-
mance of these estimators.The energy dissipation e is
taken as a test quantity.Furthermore, the in¯uence of
noise and particle concentration on the robustness of the
numerical scheme is investigated.
The numerical method was also employed to PTV data
obtained from an experiment with water ¯owing over a
forward-facing step.The energy dissipation together with
the proposed estimators was computed along a few par-
In Sect.2, the PTV arrangement used for the experi-
ments is outlined together with an analysis of expected
measurement errors.The numerical scheme is explained
in Sect.3, and its performance is tested for the case of
Burgers' vortex in Sect.4.Section 5 is devoted to a short
description of the experimental setup and the results
gained by means of the numerical method are presented.
In the ®nal section, the conclusions are given.
Experiments in Fluids 30 (2001) 492±499 Ó Springer-Verlag 2001
Received: 11 March 1999/Accepted: 8 August 2000
er (&), S.Blaser
Institute of Hydromechanics and
Water Resources Management
Swiss Federal Institute of Technology
This work was supported by a grant of the Swiss National Science
Foundation (Grant no.2-77-240-96).