NASA Technical Memorandum 86019
NASA-TM-860l9 19840024682
The
Structure
of
the
Vorticity
Field
in
Turbulent
Channel
Flow
Part
1:
Analysis
of
Instantaneous
Fields
and
Statistical
Correlations
Parviz
Moin
and
John
Kim
August 1984
NJ\S/\
National Aeronautics and
Space
Administration
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NASA Technical Memorandum 86019
The
Structure
of
the
Vorticity
Field
in
Turbulent
Channel
Flow
Part
1:
Analysis
of
Instantaneous
Fields
and
Statistical
Correlations
Parvlz
MOIn,
John Kim, Ames
Research
Center,
Moffett
Field, California
NI\SI\
National Aeronautics and
Space AdministratIOn
Ames Research Center
Moffett Field, California 94035
---
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The
structure
of
the
vorticity
field
in
turbulent
channel
How
Part
1.
Analysis
of
instantaneous
fields
and
statistical
correlations
PARVIZ
MOIN
AND
JOHN
KIM
NASA Ames Research Center, Moffett Field, California 94035
An investigation into the existence of hairpin vortices in turbulent channel
flow
is
conducted using
a database generated by the large-eddy simulation technique.
It
is
shown
that
away from the wall the distribution of the inclination angle of vorticity
vector attains its maximum
at
about
45°
to the wall. Two-point correlations of ve-
locity and vorticity fluctuations strongly support a
flow
model consisting of vortical
structures inclined
at
45°
to the wall. The instantaneous vorticity vectors plotted
in planes inclined
at
45°
show
that
the
flow
contains an appreciable number of hair-
pins. Vortex lines are used to display the three-dimensional structure of hairpins,
which are shown to be generated from deformation of transverse vortex filaments.
1.
Introduction
In 1952 Theodorsen characterized turbulent boundary layers as being composed of
large-scale horseshoe-shaped vortices which are responsible for turbulent transport.
Since then
a number of investigators have proposed physical models of turbulent
boundary layers
that
contain as their dominant feature pairs of counter-rotating
vortices with
axis either parallel or inclined to the
flow
direction. Recently, Wallace
(1982)
has collected a number of experimental results consistent with the vortex-
pair model of bc.undary layers.
He
proposes a hairpin-like vortex as the dominant
flow
structure, which is formed from the deformation, stretching, and lifting of the
transverse vortex lines.
Quantitative evidence in support of the existence of pairs of counter-rotating
vortical structures inclined to the wall and streamwise direction
was obtained from
extensive space-time correlation measurements by Willmarth & Tu (1967).
Is~cor
relation contours of the correlation between pressure fluctuations
at
a fixed point
on the wall and the spanwise velocity component,
w, in planes perpendicular to the
wall and the mean-stream direction (y-z planes) show sign reversal, with the the line
of zero correlation moving away from the wall in the downstream direction. This
result is consistent with the presence of lifting streamwise vortices which produce
reversal in
w (and hence the correlation,
pw)
across the horizontal planes containing
the vortex centers.
The
correlation between fluctuations of the streamwise velocity,
u,
at
the edge of the sublayer and the streamwise vorticity,
W~,
at
various points
above and downstream of the velocity probe was also measured. These
data
were
later
analyzed by Willmarth & Lu (1972). The location of maximum correlation
between
u and W
z
was along a line through the fixed velocity point inclined
~t
an
angle of about
10°. A quadrant analysis of the motions contributing to the
UW
z
correlation showed
that
for large negative values of u, W
z
was positive for z > 0 and
negative for
z < 0, where the u-probe was located
at
z = 0 and clockwise rotation
was denoted as positive. Based on these measurements, Willmarth
& Tu (1967)
proposed a model for the wall-layer consisting of hairpin vortices with the axis of
the primary vortex lines ( pointing in the spanwise direction when
undisturbed)
deformed, and inclined downstream and away from the wall. In their model, the
vortex lines are deformed in a regular sinusoidal manner with regions of
flow
moving
toward and away from the wall alternating in the span wise direction. This model,
with a
10°
vortex inclination angle, was proposed only for the wall region, and is
in
sharp
contrast to Theodorsen's large-scale horseshoes inclined
at
45°
to the
flow
and extending across the entire boundary layer.
Townsend
(1970, 1976) shows
that
an eddy model consisting of a pair of roller
eddies inclined
at
about 30°
to
the
flow
direction is generally consistent with two-
point correlation functions calculated from the rapid-distortion theory.
He
suggests
the double-roller eddies as the dominant structures in turbulent shears flows.
2
There is an extensive collection of experimental
data
pointing to the existence
of organized structures
or
disturbance fronts inclined to the wall and the stream
direction. Kovasnay, Kibens & Blackwelder (1970) constructed an isocorrelation
contour plot of space-time streamwise velocity correlations using a probe fixed
at
y/o
= 0.5 and another
at
different y-Iocations.
The
contour lines (in (y,t) planes)
clearly show a downstream tilt.
The
same feature is also apparent in the correlation
measurements of Blackwelder & Kovasnay (1972), where the location of the fixed
probe was close to the wall (
y/o
= 0.03 ). Tritton's (1967) two-point velocity cor-
relations decay more slowly when the probe separation line is directed downstream
away from the wall. This behavior is also evident in the correlations of Favre, Gav-
iglio
& Dumas (1957). Using space-time correlation between fluctuations of wall
shear stress and streamwise velocity with optimum time delays, Kreplin
& Eckel-
man (1979) detected a disturbance "front" which had an inclination angle of about
14°
to
the
wall
at
y+ = 50.
1
However, the front
that
can be deduced from corre-
lating spanwise fluctuations
(w,
~w
I ) does not extend beyond y+ = 30. Brown
Y
y=O
& Thomas (1977), using space-time correlation of the streamwise velocity and wall
shear stress, also detected an inclined disturbance front. However, in contrast to
Kreplin
& Eckelmann's measurements, which were limited to the wall region, they
measured the inclination angle of the front to be
18°
across the entire boundary
layer. They also propose a model in which the organized structure appears as a
horseshoe vortex.
In
contrast to Willmarth & Tu (1967) and Wallace (1982), who
describe the origin of the horseshoe vortices as due to the deformation of the pri-
mary vortex lines by random velocity and vorticity fluctuations, Brown & Thomas
attribute their origin to streamline curvature and the resulting Taylor-Gortler vor-
tices. Coles (1978) also attributes the origin of sublayer vortices to Taylor-Gortler
instability. Finally, Chen & Blackwelder(1978), in a boundary layer over a slightly-
heated wall, observed a well-defined temperature "front" across the entire boundary
IThe
superscript
+
denotes
nondimensionalizahon
with
the
wall friction velocity,
Ur
= (r/p)1/2,
and
kmematlc viscosity,
v.
3
