SPATIAL ANALYSIS OF TURBULENT FLOW FIELS BY DETERMINISTIC 
AND STOCHASTIC APPROACHES 
B. Crippa, L. Mussio 
Politecnico di Milano 
Dipartimento I.I.A.R. - Sezione di Rilevamento 
Piazza Leonardo da Vinci, 32 
20133 Milano 
ITALY 
H.G. Maas 
Institute of Geodesy and Photogrammetry 
Swiss Federal Institute of Technology 
ETH - Hoenggerberg 
CH -8093 Zurich 
ABSTRACT: 
Objects transparent to wavelenghts to which photogrammetric sensors are 
sensitive or opaque objects with 
fractal dimension near to three are 3D 
(real) examples in photogrammetric practice. The spatial analysis of 3D 
objects involves deterministic and stochasticapproaches. The former concerns 
the finite element method by using spline interpolation, the latter implies an 
optimal filtering of a signal from noise by covariance estimation, covariance 
function modelling and collocation. An application shows the discussed spatial 
analysis methods to velocity fields 
particle tracking velocimetry. 
KEY WORDS: 3D, Algorithm, Accuracy 
1. THE PROBLEM 
Three - dimensional particle tracking velocimetry 
(3D-PTV) is a well known technique for the 
determination of three-dimensional velocity fields 
in. flows. Tt is based on the discrete 
visualization of flows with small, reflecting, 
neutrally buoyand tracer particles and a 
stereoscopic recording of image sequences of the 
particles marking the flow. A powerful 3D PTV has 
been developed at the Swiss Federal Institute of 
Technology in a cooperation of the Institute of 
Geodesy and Photogrammetry with the Institute of 
Hydromechanics and Water Resources Management 
(Papantoniou/Dracos, 1989; Papantoniou/Maas, 1990; 
Maas, 1990, 1992). A flow scheme of this 3D-PTV is 
shown in Figure 1.1. 
marking of flows 
recording and digitization 
of image sequences 
v 
image preprocessing 
| image coordinate determination I hed 
7 mm 
[ establishment of correspondences i 
v 
a 3-D coordinate determination 7 
v 
| tracking in object space ] 
v 
3-D interpolation 
Figure 1.1 - Flow scheme of a 3D-PTV 
  
  
  
  
  
  
  
  
  
There are two different goals in the application 
of PTV: one is to follow a relatively small number 
of particles over a longer period of time in order 
to do Lagrangian statistics on the particles 
trajectories, the other is the determination of 
in turbulent flows determined by 3D 
There are two different goals in the application 
of PTV: one is to follow a relatively small number 
of particles over a longer period of time in order 
to do Lagrangian statistics on the particles 
trajectories, the other is the determination of 
instantaneous velocity fields from a large number 
of particles. Due to the highseeding density 
required by this second goal some ambiguity 
problems occur in the identification of particle 
images, in the establishment of stereoscopic 
correspondence (Maas, 1992b) and in tracking 
(Malik et al., 1992). It can be shown that with a 
simple stereoscopic camera arrangement positions 
of at maximum 300-400 particles can be determined 
reliably. Systems for higher spatial resolutions 
have to be based on three or even four cameras 
imaging the flow synchronously in order to be able 
to solve ambiguities in the establishment of 
stereoscopic correspondences (Maas, 1992b). 
Using standard video hardware equipment (CCIR norm 
cameras, images digitized to 512x512 pixels) a 
maximum of 1000 simultaneous velocity vectors at a 
temporal resolution of 25 velocity fields per 
second could be determined in practical 
experiments. The standard deviation of particle 
coordinates in an observation volume of about 
200x160x50 mm" determined by a three-camera system 
was 0.06 mm in X,Y and 0.18 mm in Z (depth 
coordinate). Due to imperfections of the 
calibration, illumination effects and influences 
of the shape and surface properties of particles 
their coordinates in consecutive datasets are 
correlated; it could be proved that the accuracy 
of the displacement vectors derived from the 
particle coordinates is significantly better than 
the standard deviation of particle coordinates 
(Papantoniou-Maas, 1990). 
The result of a 3D-PTV is a set of velocity 
vectors at random positions in a 3D 
observation volume which has to be interpolated 
onto a regular grid. Figure 1.2 and Figure 1.3 
show two examples of measured velocity fields.