9TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY – PIV’11
Kobe, Japan, July 21-23, 2011

Volumetric Measurement of the Transitional Backward Facing Step Flow
J. Kitzhofer1, C. Brücker1, O. Pust2, T. Nonn2
1

Institute of Mechanics and Fluid Dynamics, University of Freiberg, Germany{jens.kitzhofer,christoph.bruecker}@imfd.tu-freiberg.de
2

Dantec Dynamics A/S, Skovlunde, Denmark{oliver.pust,thomas.nonn}@dantecdynamic.com
ABSTRACTThe paper discusses the dynamical behaviour of the Backward
Facing Step Flow (BFSF) in the transitional regime at
Reh=4440. The measurements are performed via time resolved
3D Scanning Particle Tracking Velocimetry (3D SPTV)
allowing the investigation in a temporal and in a spatial
manner. The paper describes main outcomes concerning
vortex interactions, i.e. roller and hairpin vortex interaction.

mm. The aspect ratio AR (B/h) results in 57 and the expansion
ratio ER (h2/h1) results in 1.25.

1. INTRODUCTION
Flow Separation (FS) and Flow Reattachment (FR) have been
in focus of investigation for more than hundred years since thefamous studies of Ludwig Prandtl at the beginning of the 19th
century in Göttingen. Understanding the way flow separates
and reattaches on a surface is critical in engineering
applications.
First studies about separated flows are dated back to 1868
(Helmholtz) and 1869 (Kirchhoff) in case of bluff bodies.
From a theoretical, experimental and numerical point of view
much effort has been made in understanding the physicalbackground of the flow phenomena. No matter if laminar FS
or turbulent FS occurs, the FR is characterized by turbulent
flow behaviour. The basic for the analysis of such highly
fluctuating flows is the need “to separate the random processes
from the non-random processes“ as stated by [8]. This
statement implies the existence of ordered structures (nonrandom processes) in turbulence. Via qualitative flow
visualization some of the “orderly elements” began “to berecognized and described” as described by [9]. Nevertheless,three-dimensional coherent structures were hidden for
quantitative measurements due to insufficient measurement
techniques. Here, recent developments in volumetricmeasurement techniques allow the analysis of the “orderly
elements”. In fact, results concerning three-dimensional
coherent structures in a turbulent boundary layer show theimportance for investigation, because these structures seem toform the skeleton of the flow [10]. FS and FR are studied
herein in a BFSF. The BFSF is a simplified model of FS and
FR as the point of separation is spatially fixed.

Figure 1 Definition of geometry and major parameters:
B =570 mm, LE=1600 mm, Ubulk=0.52 m/s, ld,eq=92 mm,
l1=60 mm, l2=60 mm, h1=40 mm, h2=50 mm, h=10 mm,
δ=momentum thickness, xR=reattachment length
An iso-view of the experimental set-up is shown in figure 2.
The laser beam of a continuous Argon-Ion laser (8) (Coherent
Innova 70, 2 W) passes an optical lens system (9) to adjust thedesired thickness of the light sheets. A rotating mirror drum(7) reflects the laser beam into the direction of themeasurement volume and generates successively 4 parallel
light sheet planes (4). The particle images are recorded with animaging system consisting of three digital high-speed cameras(Photron APX RS) (3) and telecentric lenses (Sill Optics)(2).
The three camera system is installed above the water channel
with an angular displacement of about 0°, 45°, -45°. The fnumber of the lenses is adjusted to f# = 16 to reach a sufficientdepth of focus. A water filled prism reduces astigmatism in theimage planes (3). The side of the water channel (10) opposite
to the entrance side of the laser is covered with a lightabsorbing mat that reduces stray reflections, thus giving a
black background. The neutrally buoyant seeding particles (6)of roughly 100 μm are injected upstream of the BFS withrespect to [1]. The recorded volume is sized to about
90x60x15 mm³.

2. EXPERIMENTAL SET-UP
The BFSF is studied in a rectangular (50mm x 570mm) closed
circuit water channel (υ=10-6 m²/s) with canonical channel
geometry as sketched in figure 1. The distance between theduct entrance and the volume of interest, called entry length
LE is about 17 time the equivalent diameter ld,eq. The channel
flow is driven by gravity and the mean flow is regulated via
the hydrostatic head. All experiments were carried out for adefined height of the hydrostatic head generating a bulk
velocity upstream of the step of U∞=0.47m/s. The backward
facing step is positioned on the bottom of the channel withrespect to [1]. The step height h is chosen to 10mm. The
length of the ramp as well as the length of the plateau is 60

Figure 2. Experimental set-up of backward facing step
flow: 1) high speed cameras (3000 fps) 2) telecentriclenses 3) water filled prism 4) voluminous illumination 5)
backward facing step 6) seeding injection 7) mirror drum8) Ar-Ion laser 9) tilted mirror 10) turbulent water channel

3. INLET FLOW CONDITIONS
The inlet flow conditions (bulk velocity, boundary layer
thickness, development of velocity profile and sublayer scaled
velocity profile) are calculated via 2D Particle Image
Velocimetry. Three independently performed planarmeasurement realizations are used for the calculation of themean velocity field resulting in a total amount of 12000 vector
fields in 480s for statistical independence. The bulk velocity
Ubulk is calculated at x/h = -0.5 to Ubulk = 0.47 m/s, resulting in
Reynolds number Reh based on the step height of 4440 (at292.15 K). At x/h = -6 the flow exhibits a strong directionalchange. Here, the boundary layer separates and influences thetunnel flow. A shear layer develops, reattaches and recovers.
The length of the recovery region is calculated via thedeviation of several velocity profiles between -5<x/h<0.5 fromthe mean velocity profile at -0.5h. Figure 3 on the left shows
the influence of the angled edge, which ranges up to -2h. In
the range -2<x/h<0 the deviation is nearly constant. Thus, theflow is considered to have reached a fully developed stateagain at the backward facing step.

Figure 3. (left) Deviation of several velocity profiles
between -5h<x/h<0.5 from the mean velocity profile at 0.5h; (right) Sublayer scaled velocity profile in thesymmetry plane at x/h = -0.5
The mean velocity profile scaled with the wall shear velocity
at y/h = -0.5 is shown in figure 3 on the right. The wall shear
velocity uτ is estimated by the conventional Clauser chart
method [2]. The method assumes a universal boundary layer
profile. The wall shear velocity uτ is estimated by a leastsquares fitting to optimally fit the universal turbulent velocity
profile. The wall shear velocity results in uτ = (τW /ρ)1/2 = 25mm/s ≈ 1/20Ubulk. The universal constant values used for that
purpose are κ = 0.41 and A = 5.3. The scaled velocity profile
at x/h = -0.5 clearly indicates a turbulent flow regime.
The normalized boundary layer thickness θ*=θ/h is defined by
the momentum thickness with respect to the step height and is
calculated to 0.115 at x/h=-0.5.

4. VOLUMETRIC DATA GENERATION
4.1 Velocity field generation
The implemented algorithm for 3D Scanning Particle Tracking
Velocimetry is a hybrid PIV/PTV 4-frame method. Hybrid
PIV/PTV means that the first guess in which direction aparticle has moved is estimated via cross correlation in aninterrogation area. 4-frame means that if a particle has been
tracked for several time steps, the next position is estimated
via the kinematic history of the particle.
For that purpose, the tracking is performed in a single camera
view. First, 2D centroid positions of particles are calculated by
the gray-value weighted center of area resulting in an
uncertainty of 0.02 pix [3]. Next, positions where a particle
has moved are estimated by cross correlation or by thekinematic history of a particle. The 3D reconstruction is
performed for each two-dimensional trajectory following theinhomogeneous method for stereoscopic point reconstruction[4]. For a given two-dimensional trajectory, homologue
particles are searched in the other image planes. For that

purpose, the epipolar geometry is applied. The tolerance
around the epipolar line is chosen to be 0.5 pix with respect tothe uncertainty in 2D centroid estimation. Possible candidates
are then reconstructed in three-dimensional space. All particle
positions along the trajectory, which can unambiguously be
identified, are chosen to be part of the three-dimensional
trajectory. The remaining points are studied in a spatiotemporal manner. Here, the temporal length of the pieces of
the trajectory characterizes the quality function, nearest
neighbour, 3-frame and 4-frame quality function. The threedimensional position with the smallest deviation from thecalculated position is chosen. If the deviation is higher than acertain threshold, the 3D position is not reconstructed resulting
in a gap in the trajectory. If a candidate is identified to be partof the trajectory, the three-dimensional point is reconstructed
and the corresponding two-dimensional image points aredeleted from the database. This is repeated for each twodimensional trajectory resulting in a dataset consisting of
three-dimensional trajectories for each light-sheet. A mean
number vector density is calculated to about 42 vectors/cm³.
The post-processing includes the manipulation of the raw
trajectories. The manipulation of the raw data is theconnecting and smoothing of trajectories and the calculationof velocities and accelerations from the smoothed particle
path. The connection of trajectories is for two reasons: gaps in
the evaluated trajectories due to ambiguities and the use of ascanning illumination.
The identification of homologue trajectories between the lightsheets is possible due to the overlapping light sheets. This isdone via a simple nearest neighbour criterion in object spaceas the separation time between the single light sheets is small(1ms) resulting in small particle displacements.
Due to the scanning illumination, the particle positions need tobe corrected to represent the same instant in time with equal
separation times [5]0. The connection of the trajectories due togaps, ambiguities or scanning illumination is performed bymoving cubic spline interpolation with respect to [6]. The
uncertainty in reconstructed 3D position is estimated byanalysis of known marker positions resulting in RMSmm,cal =0.014 mm, which corresponds to RMSpix,cal ≈ 0.16 pix.
The transformation of the Lagrangian velocity field into theEulerian velocity field is evaluated by an inverse distanceinterpolation. Each single time step is interpolated on a grid of
15x90x60 mm³ with 1 mm spacing. In a next step, errorneous
vectors are deleted by a simple threshold of magnitude invelocity (0.5 m/s). The resulting velocity field is mean
averaged over a 3x3 kernel and further interpolated in a
temporal manner by a moving average of three time steps.
4.2 Extraction of shear layer
The extraction of shear layers is a crucial part in the analysis
of coherent structures as stated by [7]. Shear layers are formed
long before a vortex is born. This is especially true in the caseof the BFSF as vortical structures are built due to the shear
layer roll up. A simplified approach is used herein, which
identifies the shear layer by the analysis of the spatial
derivative δu/δz. This is possible due to the geometricalsimplicity of the backward facing step as the shear layer is
primarily dominated by shear resulting from streamwise
velocity gradients perpendicular to the bottom wall, see figure
4 on the left. Thus, the center line of the shear layer is definedby the local maximum of δu/δz for each x- and y-position. The
spatial connection of the single data points into a surface forgeometrical representation of the shear layer is performed by anearest neighbour and intensity criteria. For that purpose, arepresentative point is chosen as initial guess. The initial guess
for the position of the shear layer in region I is X/h = (0,y,1)T.

the front of the head and between the legs, thus defining a
possible detection criterion. Furthermore, the structures areusually found in vicinity of reattachment, thus the region ofinterest is located, too.

Figure 4. (left) Definition of du/dzmax-criterion; (right)
Comparison of Cr-criterion [7] (upper) and du/dzmaxcriterion (lower). The plot shows the symmetry plane at
y/h=0 and is colour coded by intensity of the single
criteria. White means a high intensity and black a low
intensity. White dots represent extracted local maxima
of the criteria showing the shape of the shear layer.
4.3 Extraction of roller
The swirling strength criterion is proposed by [11] for vortex
detection. The criterion is an extension of the Δ-criterion by
analysing the imaginary part of the eigenvalues resulting in aquantity for the swirling strength and the local plane ofswirling.
The three-dimensional center (x/h, y/h, z/h)T of the vortices insingle streamwise symmetry planes is estimated in thefollowing manner. The spanwise position (y/h) is defined by
the analyzed plane in y-direction. The height (z/h) andstreamwise position (x/h) is estimated via the analysis of theswirling strength value and following analysis of the velocity
vector field. For that purpose, each plane coded by theswirling strength criterion is analyzed for local maxima, which
are used for a first guess in the vortex center estimation. The
exact position results from the w-velocity profile in thevicinity of the local maxima. A characteristic of thedeveloping rollers is that the main axis of rotation is in
spanwise direction and that the direction of rotation isclockwise.
The spatial connection of vortex centers to a vortex center line
is performed via plane-wise connection of local maxima
following the definition that the vortex center line is theconnection of plane-wise local maxima with similar
properties. Beside the spatial position, properties are e.g.
vortex diameter, rotational velocity or rotational direction. The
principal is shown in figure 5 on the left and the experimental
result is shown in figure 5 on the right. The spanwise
displacement of the planes is 0.1h.

Figure 6. Instantaneous visualization of region ofinterest positioned about reattachment at t*=27 fromfigure 4.31 (left) gray: iso-surface of zero velocity, red:
iso-surface of w/U∞=0.12; black line: approximatedreattachment line (right) Blob a) visualized via Line
Integral Convolution (LIC) on slices through the hairpin
like vortical structure. LIC is superimposed by in plane
velocity vectors colour coded by normalized z-velocity
Figure 6 on the left shows the zero velocity iso surface and the+0.12w iso surface of an instantaneous velocity field in theregion of interest about reattachment. Four coherent areas of
upward directed velocities (blobs) are positioned about
reattachment in a certain arrangement. First of all, all blobs are
ranging out of the recirculation region, thus interacting withthe shear layer and the oncoming rollers. Blob a) and b) are
positioned upstream of reattachment and c) and d) are
positioned downstream of reattachment. Furthermore, theblobs are aligned in streamwise direction showing a linkage
between blobs and reattachment fingers. Figure 6 on the rightshows the visualization of an exemplary hairpin vortex and its
environment in the inner of the recirculation region (blob a).

5. RESULTS and DISCUSSION
5.1 Roller Formation
Originating from the shear layer instability, coherent vorticalstructures are formed with major axis of rotation in spanwise
direction. Several roll up configurations of the shear layer are
identified, i.e. parallel roll up, inclined roll up,
convex/concave roll up and piecewise roll up. The mostdominant configuration is parallel roll up, but inclined roll ups
are observed with inclination angles up to 30° with respect tothe step edge, see figure 7 on the right. The Strouhal number
for vortex formation is calculated to StVS=0.24. The center lineof the roller is identified to be wavy immediately after roller
genesis, which generates three-dimensionality immediately
downstream of the step. This is postulated to be the result of
the wavy shear layer roll up as shown in figure 7 on the left.

Figure 5. (left) Conceptional reconstruction of a threedimensional vortex tube (roller) (right) Reconstructed
vortex center lines. Vortex line in the range 0<x/h<4.
Streamwise planes are colour coded by swirling strength
4.4 Extraction of bursts
The measurement inside the recirculation region is rather
difficult. The flow is highly three-dimensional and unsteadywith a large amount of spatial and temporal length scales.
Nevertheless, previous investigations observed vorticalstructures inside the recirculation region, e.g. hairpin-likevortex structures. While the rollers impinge on the wall,
hairpin like vortices are generated in both flow directions.
Characteristic of hairpin like vortical structures is the lift-up
effect and the corresponding large wall normal velocities at

Figure 7. (left) Interaction of shear layer (surface) andvortex center line (black line) in the range 0<x/h<4.
(right) Probability density function of angle of roller
genesis5.2 Roller Convection
Characteristic of the rollers is the distorted vortex center line.
Here, amplitude as well as wave length is estimated. The roller
waviness showed distinct specifications depending on thestreamwise distance. Thus, the roller performs a stretching instream-wise direction in between 0<x/h<3. At x/h=3, the roller
loops are primarily oriented in stream-wise direction. Further
downstream, the rollers perform a tumbling motion resulting
in a re-orientation of the loops in a vertical alignment andfollowing tilting to a somehow streamwise alignment.

alignment in a vertical alignment and following tilting to asomehow streamwise alignment.

Figure 8. Vortex center line properties as a function of
streamwise distance (left) properties in y-z-plane (right)
properties in y-x plane
Reasons for the tumbling motion are found in the pairwise
vortex interaction of generated bursts and oncoming rollers asshown in figure 9. The figure on the top shows a measured
event, where a burst interacts with an oncoming roller. The
schematic representation is shown in figure 9 on the bottom.
One identical roller visualized via the vortex center line istracked for the sequence while influenced by the structure(t0*≈0, t1*≈2 t2*≈4, t3*≈6, t4*≈8). The hairpin like structure isvisualized via the isosurface of 0.12w/U∞ at t*≈4.5.
Furthermore, a slice at z/h=0.1 is colour coded with backward(blue) and forward (white) flow direction identifying thereattachment position at t4*. The chosen perspective shows thedeformation of the roller in vertical direction from t1* to t3*
while passing over the upward directed velocity. From t3* tot4* the vertical aligned vortex loop is reoriented to an
inclination angle of roughly 30° with respect to the bottom
wall. In between t2* and t4*, the distance between vortex andbottom wall is rather large and consequently results in local
large amount of backflow generating the backflow finger.

6. CONCLUSION
This article discusses the dynamical and spatial behaviour of
the BFSF. In detail, we focus on the generation of the rollers
immediately downstream of the step and on the spatial
characteristics of the rollers while convecting downstream.
Several roll up configurations of the shear layer are identified,i.e. parallel roll up, inclined roll up, convex/concave roll up
and piecewise roll up. The most dominant configuration isparallel roll up, but inclined roll ups are observed withinclination angles up to 30° with respect to the step edge. The
Strouhal number for vortex formation is calculated to
StVS=0.24. The center line of the roller is identified to be wavy
immediately after roller genesis generating threedimensionality downstream of the step. This is postulated tobe the result of the roll up of the wavy shear layer. The roller
waviness showed distinct specifications depending on thestreamwise distance. A dominant tumbling motion is observedin vicinity of reattachment, which is the result of vortex pairinteraction. A roller locally generates larger backflow
resulting in a jet directing into the recirculation region. Due tothe jet, a hairpin like vortex is born in the inner of therecirculation region. The hairpin vortex interacts with the next
oncoming roller and acts as a local bump. The interaction
results in a re-orientation of the roller loops from a streamwise

Figure 9. (top) Measured hairpin vortex - roller vortex
interaction in the vicinity of reattachment (bottom)
Schematic of roller development as a function of
streamwise distance

REFERENCES[1] Beaudoin J.-F., Cadot O., Aider J.-L., Wesfreid J.E.
(2004) Three-dimensional flow over a backward-facing
step. Europ. Journal of Mechanics B/Fluids 23, 147-155.
[2] Clauser F. (1956) The turbulent boundary layer. Adv.
ApplMech 4, pp.1.51[3] Ponitz B., (2010) Experimentelle Untersuchungen derturbulenten Stufenumströmung in einem Wasserkanal
mittels
drei-dimensionaler
optischer
Strömungsmesstechnik, Diplom Thesis, TU Freiberg[4] Schreer O. (2005) Stereoanalyse und Bildsynthese,
Springer, Berlin[5] Hoyer K, Holzner M., Lüthi B., Guala M., Liberzon A.,
Kinzelbach W. (2005) 3D scanning particle tracking
velocimetry Exp. Fluids 29: 145-53[6] Lüthi B.,Tsinoba A., Kinzelbach W. (2005) Lagrangianmeasurement of vorticity dynamics in turbulent flow. J.
Fluid Mech. 528: 87-118[7] Schafhitzel T., Baysal K., Vaaraniemi M., Rist U.,
Weiskopf D. (2010) Visualizing the Evolution andInteraction of Vortices and Shear Layers in TimeDependent 3D Flow. To appear in IEEE Transaction on
Visualization and Computer Graphics[8] Dryden, Hugh L. Recent advances in the mechanics ofboundary layer flow. Advances in Applied Mechanics 1,
Academic Press, New York, pp.1-40[9] Roshko A. (1976) Structure of Turbulent shear flows: A
new look, AIAA Jpurnal, vol. 14, no. 10, pp. 1349-1357[10] Schröder A. Geisler R., Staack K., Elsinga G.E., Scarano
F., Wieneke B., Henning A., Poelma C., Westerweel J.
(2010) Eulerian and Lagrangian views of a turbulent
boundary layer flow using time-resolved tomographic
PIV. Exp. Fluids 50: 1071-1091 pp.
[11] Zhou J., Adrian R.J., Balachandar S., Kendall T.M.
(1999) Mechanism for generating coherent packets ofhairpin vortices in channel flow. J. Fluid Mech. 387: pp.
353-396