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TURBULENT COHERENT STRUCTURES IN A THERMALLY STABLE
BOUNDARY LAYER
Owen Williams and Alexander J. Smits
Department of Mechanical and Aerospace Engineering
Princeton University
Princeton, NJ 08540
owilliam@princeton.edu
ABSTRACT
The effects of thermal stability on coherent structures in
turbulent flat plate boundary layers are examined experimen-
tally. Thermocouple and DPIV measurements are reported
over a Richardson number range 0 <Ri
δ
<0.2. The reduc-
tion in wall shear and the damping of the turbulent stresses
with increasing stability are qualitatively similar to that found
by Ohya et al. (1996) including the major changes observed
when the flow enters the strongly stable regime. In contrast, a
critical bulk Richardson number of 0.05 is observed,which is
much lower than the value of 0.25 found in this earlier study.
In the weakly stable regime, hairpin vortices are seen to con-
tinue to populate the near-wall region and are elongated in the
streamwise direction creating a smaller angle of inclination to
the wall. With increasing stability, the angle of these struc-
tures continues to decrease and they are confined closer to the
wall. In our experiments, the strongly stable flows show no
evidence of large scale structures, or the presence of gravity
waves.
INTRODUCTION
Thermally stable boundary layers are commonly found
in arctic regions above the ice pack where the ice is typically
at a lower temperature than the air flowing over it. Thermal
stability causes a severe reduction in the turbulent fluxes and
the heat transfer from the surface. Current General Circula-
tion Models (GCM) are usually based on a form of Monin-
Obukhov similarity theory, where the atmospheric surface
layer is assumed to have either a constant vertical heat flux,
or a modified form called local-scaling that uses a local heat
flux. The vertical extent over which these theories are valid
shrinks with increasing stability such that parameterizations
based on them need to be significantly modified at stronger
stratifications (Mahrt, 1998). For example, King et al. (2001)
compared four such parameterizations in a coarse mesh simu-
lation of the atmosphere over the Antarctic. They found a total
surface heat flux variation of over 20 W/m2among the mod-
els, corresponding to surface average temperature differences
of greater than 10◦C, indicating that there are still significant
gaps in our understanding of these flows.
Stable boundary layers are generally classified as ei-
ther weakly stable, corresponding to a nocturnal atmospheric
boundary layer at moderate latitudes for which Monin-
Obukhov similarity is valid, or strongly stable, represented
by the arctic boundary layer for which current models are in-
sufficient. Mahrt (1998, 1999) describes some of the impor-
tant differences between these two regimes, including the in-
creasing prominence of gravity waves, meandering motions,
intermittency, increased anisotropy and the possible detach-
ment of turbulence from the surface with intermittent recou-
pling. Gravity waves are believed to explain the existence of
a counter-gradient flux sometimes observed at higher strati-
fications (Thorpe, 1972). Mahrt (1998) notes that a single
definition of the strongly stable regime remains controversial
and elusive since all of these phenomena are rarely observed
within the same study. A better understanding of the strongly
stable regime also hampered by measurement difficulties be-
cause small fluxes necessitate better instrumentation and sig-
nificantly longer averaging times.
Of particular interest is the possible existence of a critical
stratification that describes the transition between the strongly
and weakly stable regimes. There are many parameters that
are used to describe the extent of thermal stratification but it
is currently unclear which parameter is the most appropriate.
Apart from the Monin-Obukhov length, the most commonly
cited parameter is the gradient Richardson number:
Ri =g
Θ
∂
Θ/
∂
z
(
∂
U/
∂
z)2(1)
which describes the relative influence of the stabilizing effect
of buoyancy and the destabilizing effect of shear. Here, Θ
is the potential temperature, Uis the mean velocity, zis the
wall-normal distance and gis the gravitational constant.
It has been shown that turbulent statistics such as stream-
wise intensity and Reynolds shear stress correlate well with
this quantity (Arya 1974; Ohya et al. 1996). In addition, it
was established by Miles (1961) and Howard (1961) that a
1
Figure 1. PIV setup to measure turbulent statistics of a thermally stable boundary layer developing on the underside of a heated
plate
laminar, steady, inviscid flow will remain stable to small per-
turbations if Ri >0.25 everywhere. This is a sufficient condi-
tion that was first predicted by Taylor (1931) and later verified
experimentally by Scotti and Corcos (1971). This criterion
has since been extended to compressible flows by Chimonas
(1970) giving the same result. It should be noted, however,
that unsteadiness in these flows has been shown cause insta-
bility at Richardson numbers greater than 0.25 (Majda and
Shefter, 1998), possibly helping to explain some of the vari-
ability in the atmospheric data due to its natural transience.
Here, due to the limits of our experiment, we will primarily
consider the bulk Richardson number,
Ri
δ
=g
δ
Θ∞
ToU2
∞
(2)
which is similar to the gradient Richardson number but where
the gradients are evaluated across the entire layer.
θ
∞is the
temperature difference across the layer, Tois the average ab-
solute temperature, U∞is the freestream velocity, and
δ
is the
boundary layer thickness.
Although the condition Ri >0.25 has been shown to be a
sufficient condition for the maintenance of laminar flow un-
der certain conditions, it does not necessarily apply to the
cessation of turbulence within an already turbulent flow. The
first analysis of turbulent stratified flows by Richardson (1920)
predicted that for Ri >1 no turbulence would survive. Viscous
dissipation was neglected in this analysis and this has since
been found to be important for strongly stable flows. Recent
experimental and observational studies have indicated, how-
ever, that this criterion is actually more robust than initially
anticipated because turbulence actually has been observed to
exist for Ri >> 1 (Galperin et al. 2007). Additionally, models
of stratified turbulence that use a critical Richardson number
as a threshold for the extinction of turbulence have been found
to have insufficient mixing if the critical Richardson number
Ric<1 (see Galperin et al. 2007 for discussion). Recent work
by Canuto (2001) showed that the presence of radiative losses
and internal gravity waves acts to reduce stratification, further
increasing the Richardson number required for the suppres-
sion of turbulent mixing. Strong stratification has also been
observed to increase anisotropy and horizontal mixing even
when vertical mixing has been largely suppressed. This ob-
servation leads Galperin et al. (2007) to conclude that a single
critical Richardson number for the suppression of turbulence
does not not exist.
Other works have used a flux Richardson number (Rif),
defined as the ratio of work done against buoyant forces to
the production of turbulent, two terms in the turbulent kinetic
energy equation. That is,
Rif=g
θ
w
Θ
uw
∂
U
∂
z
(3)
Here,
θ
wis the turbulent heat flux, Θis the local average tem-
perature and uw is the Reynolds stress.
While the full problem of reverse transition due to strat-
ification is presently intractable, simplified analyses based on
equations of turbulent kinetic energy, mean square temper-
ature fluctuations, and turbulent heat flux have been devel-
oped. Ellison (1957) first used this approach, modeling the
dissipation terms as the ratio of the particular quantity to its
decay time. Defining the critical stratification as that corre-
sponding to a condition where continuous turbulence cannot
be maintained, he arrived at a critical Richardson number of
Rif=0.15. A following study by Townsend (1958) based his
model on an expected variation in turbulent Prandtl number,
and suggested the threshold Rif=0.5. Ayra (1972) improved
on this approach with measured values, and found a critical
value Rif=0.15 −0.25. These analyses allow for the fact
that above this critical value intermittent turbulence can oc-
cur: the flux Richardson number is a local quantity that for a
given flow can fluctuate above and below the critical stratifi-
cation level.
It is difficult to match these critical Richardson num-
ber estimates with atmospheric observations as they are lo-
cal quantities and the definitions of weak and strong stabil-
ity are more macroscopic in nature. Additionally, it is un-
clear whether alternative global parameters such as the bulk
Richardson number are sufficient to characterize the differ-
ences between these weakly and strongly stable flows.
There are only a limited number of previous laboratory
experiments that have examined the effects of thermal stabil-
ity on turbulence. It was found by Ayra (1974) (Ri
δ
<0.1)
2
Experimental Case NV4 T1V4 T2V4 T3V4 T4V4 T5V4 T6V4 T7V4
∆Θs0 19.7 42.8 60.4 76.3 100.0 115.5 128.9
Re
θ
1032 986 1048 990 901 907 813 773
Ri
δ
0 0.0087 0.035 0.048 0.060 0.077 0.088 0.097
u
τ
/U∞0.048 - - - - - - -
Table 1. Flow properties with varying wall temperature but constant velocity of U∞=1.44m/s
Experimental Case T3V1 T3V2 T3V3 T3V4 T3V5 T3V6 T3V7 T3V8 T3V9
U∞(m/s)0.96 1.12 1.28 1.43 1.61 1.78 2.08 2.42 2.63
Re
θ
696 774.8 841 990 1169 1370 1533 1803 2040
Ri
δ
0.11 0.079 0.061 0.048 0.038 0.031 0.023 0.017 0.014
Table 2. Flow properties with varying velocity but constant ∆Θs=60
and also Ogawa et al. (1985) (Ri
δ
<0.25)that, unlike un-
stable flows, the mean velocity profile shows significant devi-
ations from the neutral case even at moderate stratifications.
They also observed a marked reduction in turbulent intensity
and fluxes with increasing Richardson number. Further in-
vestigations by Ohya et al. (1996) (Ri
δ
<1.33)found only
gradual deviation of turbulent quantities from the neutral case
for weak stratification, but significant reductions in turbulent
intensity were found at stronger stratifications. Furthermore,
the near-wall turbulence peak moved away from the wall with
increasing stability. A critical stratification, Ri
δ
=0.25, was
found to separate these two regimes, a value that agrees with
the analysis of Miles-Howard theory for laminar flow or Arya
(1972) for the onset of intermittancy.
These experiments were limited to single point mea-
surements of velocity and temperature, and little information
on the behavior of the turbulent structure is currently avail-
able. Here, we investigate the differences between weakly and
strongly stable flows, and examine the changes in the coher-
ent structures, such as the nature of the hairpin vortices, with
increasing stability. Strongly stable flows were also examined
for the presence buoyantly driven structures such as gravity
waves.
EXPERIMENTAL APPARATUS
The experiments were conducted in 5 m long, 1.2 m by
0.9 m cross-section, open-return wind tunnel that was modi-
fied to study thermally stratified flows. The tunnel was oper-
ated at freestream velocities between 0.8≤Ue≤2.5m/s. The
upper surface of the tunnel was replaced with a 12.7 mm thick
aluminum plate backed with strips of heating tape allowing
the plate to be heated isothermally. Eight thermocouples were
mounted on the centerline of the plate to ensure that this con-
dition was maintained. The freestream temperature was also
measured using a thermocouple. The flow was tripped using
a 6.35 mm rod mounted to the leading edge of the plate, just
after the convergent section of the tunnel. The experimental
apparatus is shown in Figure 1.
The experiment was conducted at nine velocities (V1-
V9) for each of eight wall temperatures. Including the neu-
trally stable case, they were labelled N, S1–S7. The tem-
perature difference between the wall and freestream, ∆Θs
varied between zero (N) and 130◦C(S7). The correspond-
ing Richardson number and Reynolds number ranges were
0≤Ri
δ
≤0.2 and 600 ≤Re
θ
≤2050.
Particle Image Velocimetry (PIV) was used determine
the velocity field in a plane containing the wall-normal and
streamwise directions. A New Wave Tempest and Gemini
dual head ND:YAG laser system was used as the laser source.
Each laser delivers 100 mJ energy per pulse at a wavelength of
532 nm. The flow was imaged with a PCO.1600 Camera with
an interframe time of 300
µ
s. Seeding was generated using an
MGD Max 3000 APS mineral oil based fog generator. It was
injected into a large enclosure attached to the inlet of the tun-
nel allowing the particles to be well-mixed with the incoming
air before entering the tunnel inlet.
The PIV images were processed using the a modified
WIDIM code detailed in Scarano and Riethmuller (2000). It is
an adaptive multigrid scheme that uses iterative image defor-
mation to enhance correlation and reduce peak-locking. The
final window size was 32×32 pixels with 50% window over-
lap. The regression filter was set to 2. The internal signal to
noise filter was disabled because it was found to have negli-
gible impact on statistical resuls while requiring a large pro-
portion of vectors to be interpolated. In some higher stability
cases, near-wall seeding was found to be insufficient due to the
strong local density gradient and low levels of mixing. These
regions were cropped from the images and results.
The field of view of PIV measurements was approxi-
mately half the boundary layer height so the remaining mean
velocity profile was measured using a Pitot tube. The static
3
pressure was measured using a static pressure probe mounted
in the freestream. An Omega PX653-0.05BD5V high accu-
racy, pressure transducer was used. Using these profiles, the
boundary layer thickness and freestream velocity could be
estimated. Due to the low dynamic pressures involved and
the variation in density across the layer, the Pitot tube pro-
files measured at higher wall temperatures were found to be
unreliable. Therefore, the boundary layer thicknesses and
freestream velocities found in the neutral case were used to
non-dimensionalize the data for the stable cases, as well.
RESULTS AND DISCUSSION
As data was taken varying both temperature and veloc-
ity, two sets of statistics will be shown, keeping one of these
variables constant. The case with constant velocity enables us
to examine the statistics with the smallest Reynolds number
variation. Other data sets show very similar trends and are not
included. Tables 1 and 2 list the global properties of each of
these flows.
The mean velocity profiles, shown in Figure 2(a), show a
strong reduction in wall shear as the increasing level of stabil-
ity decreases turbulent mixing. The strongest stability cases
are almost laminar in nature. These profiles are qualitatively
very similar to those shown by Ayra (1974) and Ohya et al.
(1996) .
The damping of turbulence is clearly seen in Figures 2(b)
and 2(c), and the data are in good qualitative agreement with
the results obtained by Ohya et al. (1996) . As with pre-
vious studies, the profiles can be divided into two regimes:
the weakly stable, with minor reductions in turbulence inten-
sity and shear stress, and the strongly stable where the tur-
bulent stresses are significantly damped. The strongly stable
stable profiles are also observed have a fundamentally differ-
ent shape, with the peak in turbulence intensity moving away
from the wall. This phenomena was also observed in Ohya
et al. (1996). Case T3V4, common to both figures, appears
to represent a transitional state between these two regimes and
we will refer to it as the critical case. One of the most interest-
ing aspects of our results was the critical Richardson number.
It was found to be Ri
δ
=0.05, which is much lower than the
critical values measured by Ohya et al. (1996) or predicted by
Miles-Howard theory (Miles, 1961) or Arya (1972).
To examine whether this discrepancy is a Reynolds num-
ber effect, Figure 3 plots Re
θ
against Ri
δ
for all the cases
studied in our experiment. The data were divided into weakly
and strongly stable categories based on the behavior of the tur-
bulent intensity profiles. While the Reynolds number range
near the critical value is small in extent, it can be seen that
the value of 0.05 defines a clear threshold above which the
flow becomes strongly stable. Although Ohya et al (1996)
do not quote momentum thickness Reynolds numbers, they
were estimated from mean velocity profiles to be in the range
2500 ≤Re
θ
≤5000 and thus it seems unlikely that the differ-
ences between these two critical Richardson numbers can be
ascribed to Reynolds number differences.
The source of the discrepancy in critical Richardson
number is unclear. The statistics show a finite turbu-
lence intensity within the strongly stable regime whereas the
Reynolds stress is almost identically zero. It is possible
that the remaining turbulence is uncorrelated noise and the
600 800 1000 1200 1400 1600 1800 2000 2200
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Reθ
Riδ
Weakly Stable
Strongly Stable
Figure 3. Reynolds and Richardson numbers for all cases
studied here.
Figure 4. Structure of boundary layer for T3V4 (Ri
δ
=
0.048). Symbols as in Figure .
flow is in fact fully relaminarized. Alternatively, the peak
in Reynolds stress observed by Ohya et al. (1996) occurred
within the outer layer of the boundary layer and it is pos-
sible that this was missed in our study due to the restricted
field of view. In addition, it is possible that these flows are
more sensitive to initial conditions than previously thought
since Ri <0.25 is not large enough to force the laminar flow
to remain laminar when tripped. Further studies will investi-
gate the effect of trip wire size and wall roughness on critical
Richardson number.
The PIV data were then used to examine the changes
in turbulent structure between the weakly and strongly stable
cases. Structure was examined by plotting velocity vectors
and contours of constant swirl criteria over vorticity fields, as
recommended by Adrian et al. (2000).
Representative results for the neutral, weakly stable and
strongly stable regimes are shown in Figure 5 . Hairpin vor-
tices with an inclination angle of approximately 45◦were seen
in the neutrally stable case and what appeared to be “older”
hairpins were observed further away from the wall, in line
with the model proposed by Adrian (2007). These older struc-
4
tures were not observed in the weakly stable regime, and the
near-wall hairpins were stretched in the downstream direc-
tion, apparently in response to the additional work necessary
to overcome the vertical density gradient. This characteristic
structure angle was observed to continue to decrease with Ri
δ
.
Figure 5(c) shows that once the flow had progressed to
the strongly stable state, all large scale turbulent structure has
been suppressed. Additionally, no gravity waves were ob-
served. This is consistent with the observed reduction in tur-
bulence intensity.
A snapshot of the critical case (T3V4) is shown in Figure
4. The structure is a combination of the weakly and strongly
stable regimes, with stretched hairpin vortices of significantly
reduced strength appearing intermittently. This intermittency
is most likely connected with fluctuations in the local Richard-
son number variation. Conventionally, it is thought that as tur-
bulence is reduced, shear begins to build up due to increased
stratification. This should then trigger a shear instability such
as Kelvin-Helmoltz waves which increase mixing, reducing
stratification until they are then damped out. No strong grav-
ity waves were observed within this experiment, possibly in-
creasing the sharpness of the transition between weakly and
strongly stable flows.
CONCLUSIONS
Mean and fluctuating turbulent statistics were measured
within a thermally stable boundary layer using PIV. The wall
shear was found to reduce significantly with increasing stabil-
ity and mean velocity profiles approached the laminar case.
Turbulent intensities and stresses could be separated into
weakly stable and strongly stable regimes. These results were
found to be qualitatively similar to the studies of Arya (1974)
and Ohya et al. (1996), although the critical stratification
between the two regimes was Ri
δ
=0.05, which is signifi-
cantly lower than that observed by Ohya et al. (1996). Within
the weakly stable regime hairpin structures were observed to
remain confined to the near-wall region and were elongated
in the streamwise direction when compared with the neutral
case. The angle of these structure was observed to continue
to decrease with increasing stratification. Large-scale struc-
ture was found to have been damped within the strongly sta-
ble regime and no gravity waves were observed. As gravity
waves are one mechanism that can increase local mixing, it
is thought that their absence helped contribute to the sharp-
ness of the observed transition to a strongly stable state. At
critical stratification, hairpin vortices with a shallow angle to
the freestream were intermittently observed in a flow that was
otherwise strongly stable in nature.
ACKNOWLEDGEMENTS
We would like to thank Princeton University’s Grand
Challenges–Energy program and the Thomas and Stacey
Siebel Foundation for funding this research.
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5
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
U/U∞
Z/δ
T3V2
T3V4
T3V6
T3V7
T3V8
T3V9
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
U/U∞
Z/δ
NV4
T1V4
T2V4
T3V4
T4V4
T5V4
T6V4
T7V4
(a) Mean velocity profiles for varying freestream velocities (left), and varying wall temperatures (right)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
u′/U∞
Z/δ
T3V2
T3V4
T3V6
T3V7
T3V8
T3V9
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
u′/U∞
Z/δ
NV4
T1V4
T2V4
T3V4
T4V4
T5V4
T6V4
T7V4
(b) Streamwise turbulent intensity profiles with varying velocity (left) and varying wall temperature (right)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10−3
0
0.1
0.2
0.3
0.4
0.5
0.6
−u′ v′/U∞
2
Z/δ
T3V2
T3V4
T3V6
T3V7
T3V8
T3V9
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10−3
0
0.1
0.2
0.3
0.4
0.5
0.6
−u′ v′/U∞
2
Z/δ
NV4
T1V4
T2V4
T3V4
T4V4
T5V4
T6V4
T7V4
(c) Reynolds stress profiles with varying velocity (left) and varying wall temperature (right)
Figure 2. Variation in turbulent statistics with velocity (left) and wall temperature (right).
6
(a) Neutrally stable boundary layer
(b) Weakly stable (Ri
δ
= 0.014)
(c) Strongly stable (Ri
δ
= 0.08)
Figure 5. Turbulent structures within stratified and neutrally stable boundary layers. These are visualized using a combination of
vorticity contours, swirl strength and a Galilean transform of the velocity field.
7
