12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

EXPERIMENTAL STUDY ON A TRANSITION OF FLOW IN WEAKLYUNSTABLY STRATIFIED TURBULENT BOUNDARY LAYERS

Yoshinori Mizuno∗ , Toshimasa Yagi and Kazuyasu Mori
Meteorological Research InstituteNagamine 1-1, Tsukuba, Ibaraki, Japan∗ ymizuno@mri-jma.go.jp

ABSTRACTof the system size. While it is still difficult for direct numericalsimulations to achieve high Reynolds numbers encountered inengineering and nature, laboratory experiments have a relativeadvantage to attain a higher Reynolds number and to acquirelong-time measurements. We perform an experimental study
on the transition of flow structures caused by the rise of buoyancy effect in weakly stratified boundary layers.

We investigate the dependence of turbulence statistics andstructures in weakly unstably stratified turbulent boundary layers through wind-tunnel experiments. In the constant-flux
layer where some universal properties of turbulence such asa similarity of turbulent statistics are expected to be observed,
the production rate of turbulent kinetic energy due to shear increases at large scales comparable to the boundary layer thickness δ with the relative strength of buoyancy effect, evaluatedby −δ /L where L is the Obukhov length. It is found that thevertical profile of the variance and spectrum of the streamwise
velocity component are not modified by the stratification up to−δ /L ≈ 1. On the other hand, they exhibit a drastic change
beyond −δ /L = 1, that is likely to be attributed to a transition in the flow structures due to the buoyancy effect. The
two-point correlation functions of horizontal velocity fluctuations indicate the emergence of roll-like vortical structures for−δ /L > 1.

EXPERIMENTS
The data set for the present analyses was generated fromexperiments in an open-return wind tunnel at MeteorologicalResearch Institute, located at Tsukuba, Japan. The test section is a uniform rectangular duct, which is 18 m long, 3 mwide, and 2 m high. Aluminium rods with a diameter of 3
mm were placed at intervals of 100 mm on the floor to create rough surface throughout the test section. The rectangularcross-section with the same dimensions extends 4 m upstream
from the test section, and the rods were also placed within thispart in the same manner. Measurements were performed 14 m
downstream from the entrance of the test section by a laserDoppler velocimeter (LDV, Dantec model F60 with 60X17).
Its sensing volume has a diameter of 120 µm and length of
1.5 mm oriented in the spanwise direction. A boundary layer
was formed by blowing air at a constant speed into the test section, and the streamwise and wall-normal velocities were measured at different heights along a traverse through the boundary
layer. Since the signals of the two velocity components wererecorded independently, their time-sequences were resampled
for synchronization to evaluate their cross-correlation.
Unstable stratification was generated by heating the floorof the test section. The temperature of the floor was controlled
so that its difference from that of incoming air is constant. The
thermal flux Q0 on the floor was measured by two plate-type
heat flow sensors (Eto-denki, M55A). Each has the dimensionsof 50 × 50 × 0.7 mm. On the floor, they are mounted up- anddownstream sides of an aluminium rod under the LDV probe.
The boundary layers discussed in this paper are summarizedin Table 1. Figure 1 shows two examples of visualization ofneutral (N) and unstable (U3) boundary layers. In the unstablecase in 1(b), we can recognize some tall vertical fluid motions
which is absent of the neutral case in 1(a). We will show how
turbulence statistics are modified by the emergence of suchbuoyancy-driven motions.
We also obtained two-dimensional velocity fields in a horizontal plane by a Particle Image Velocimetry (PIV) technique(Flowmap, Dantec Dynamics A/S). The measurement field ofview is 496 mm long and 397 mm wide for all the cases. Statistics for each case are compiled from 9000 (N), 6600 (U2) or

INTRODUCTION
In an unstably stratified boundary layer, both shear andbuoyancy effects contribute to the production of turbulent kinetic energy. According to a dimensional analysis (Monin &Obukhov, 1954), the turbulence statistics measured at the wallheight z is parameterized by z/L, where L ≡ −u3∗ Θ/κgQ0 is
the Obukhov length, and u∗ is the friction velocity, Q0 is thetemperature flux on the wall, g is the gravitational acceleration,and Θ is a representative temperature in the boundary layer.
Since Q0 is defined to be positive when the flux is upward, L
is negative for unstably stratified cases. The absolute value |L|
is interpreted as the height at which the effect of buoyancy becomes important (Wyngaard, 2010). The ratio z/|L| measuresthe local strength of stratification at the height z, while δ /L
where δ is the boundary layer thickness measures the global
strength of stratification throughout the boundary layer.
It is widely known that streamwise rolls spanning almost
full height of the boundary layer appear under weakly unstablestratification (Etling & Brown, 1993). Due to the presence ofsuch large-scale rolls, streamwisely long stripes of upward anddownward motions along the rolls are observed, and they are
responsible for the vertical transfer of momentum, heat and soon. Detailed and systematic studies on the structures in a turbulent boundary layer parameterized by the stability parameterδ /L or z/L are rarely found. Among them, a recent direct numerical simulation (Pirozzoli et al., 2017) of a horizontal channel under unstable stratification showed that streamwise rolls
appear for −δ /L up to O(1). Blass et al. (2020) also pointed
out that the buoyancy effect emerges after L excesses a factor

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

(b)

Figure 1. Turbulent boundary layers (a) neutrally stable and (b) unstable at −δ /L = 2.6 visualized by smoke-wire in a wind tunnel.
The air flows from right to left.

2300 (U3) images obtained at the sampling rate of 0.5 Hz.
Hereafter the streamwise, spanwise and wall-normal (vertical) coordinates are denoted by (x, y, z), and velocity fluctuations by (u, v, w) in these directions, and an overline stands foraveraging in time.

(Townsend, 1976; Marusic et al., 2013). As in the figure, thevariance does not show any clear dependence on the stability
parameter up to −δ /L ≈ 1, but the most unstable case (U3)obviously deviates from the other cases, implying a transitionof the flow by increasing the stability parameter −δ /L beyond−δ /L ≈ 1. In that case, the variance u2 and covariance uw(and also w2 though not shown here) have the second peak intheir magnitude in the outer layer, at z/δ ≈ 0.5. At this location, corresponding to −z/L ≈ 1, the buoyancy effect explicitly emerges in those profiles.
The vertical momentum flux uw and also production-rateof turbulent kinetic energy −uwdU/dz are enhanced by unstable stratification at large scales comparable to δ , as shown infigure 3(a), but gradually even for −δ /L < 1 where variances
are not modified. The peak is located at λ /δ ≈ 5 for all theunstable cases, and it seems not dependent of the height z neither. The enhancement of momentum flux is observed only atlarge scales (Mizuno et al., 2022), but it does not modify the
structure of u for −δ /L. As expected from the behavior of thevariance, the spectrum of u shown in figure 3(b) is robust up
to −δ /L ≈ 1, and again drastic change occurs when −δ /L isfurther increased. The mechanism of energy transfer at least
for the streamwise velocity component in the neutral case isretained up to −δ /L ≈ 1, and the buoyancy effect contributes
only to increasing the energy production and does not affectthe flow structures.
To examine the buoyancy effect on flow structures, a large
amount of snapshots of the two-dimensinal velocity field on ahorizontal plane obtained by PIV are compiled to the two-point
correlation function of velocity fluctuations, defined by

Table 1. Parameters in the present boundary layers. Here U∞
is the free-stream velocity, ∆Θ is the temperature difference
between the incoming air and floor, and Reτ is the friction
Reynolds number.

U∞ [m/s]∆Θ [K]

Reτ−δ /L

Symbol

N

1.5

-

2380

–

×

U1

3

20

4620

0.43

N

U2

2

20

3760

1.2

u

U3

1.5

20

3810

2.6

u

RESULTS
The profiles of some one-point statistics in the present
boundarylayers are shown in figure 2. We here use u∗ =√−uw within the vertical range where the momentum flux isconstant, so called constant-flux layer, as a velocity scale. This
gives a good approximation for the friction velocity, though
not discussed in detail here. For all the boundary layers realized in the present experiments, the layer is indeed found up toz/δ ≈ 0.2, as shown in figure 2(b).
Figure 2(a) compares the mean streamwise velocity U for
different values of δ /L. The profiles are expected to collapse
to the neutral case near the wall where the effect of shear
is dominant against the stratification, and start diverging as
the height z increases. The effect of stratification apparently
emerges gradually on the mean velocity profile as −δ /L is increased.
The profiles of the variance u2 and covariance uw are
shown in figure 2(b). For the neutrally stable case, a logarithmic dependence of the variance u2 of the streamwise velocity fluctuation was predicted and supported by experimentsα(x + rx , y + ry )α(x, y)q,Cαα (rx .ry ) ≡ qα(x + rx , y + ry )2 α(x, y)2
where α = u or v. Though the length of the measurement fieldof view is comparable, or even shorter than δ , it is confirmedthat the functions for the neutral case N agree well with a DNS
by Sillero et al. (2014), at least, for rx and ry within the rangesof the field of view. The effect of unstable stratification on thecorrelation functions are here presented at z/δ ≈ 0.15. Figure4(a) shows that u has a longer correlation length in the streamwise direction for the unstable cases U2 and U3, which seems
to be attributed to the emergence of roll-like vortical structuresin these cases. For the neutral case, the streamwise velocity

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 2. Vertical profiles of (a) the mean streamwise velocity U and (b) Reynolds stresses u2 and uw. Symbols are as in Table 1. The
solid lines are the profiles in a DNS of a neutrally stable case for Reτ = 1999 by Sillero et al. (2013).

Figure 3. (a) Cospectrum of u and w and (b) spectrum of u as functions of the streamwise wavelength λ at z/δ = 0.12 (solid) and
0.18 (dashed). Symbols are as in Table 1.

−δ /L = 1, that is likely to be attributed to a transition in theflow structures due to the buoyancy effect on the flow. The
two-point correlation functions of horizontal velocity fluctuations also indicate the emergence of roll-like vortical structuresfor −δ /L > 1.
Our present measurements can extract only a portion offull information on the flow, that is not enough for comprehensive understanding of flow structures in stratified boundary
layers. In addition, the present range of δ /L probably reaches
only the beginning of the transition toward the buoyancy dominat regime (Blass et al., 2020). It is left as a future work toachieve further unstable stratification experimentally.

fluctuations at two separate locations in the spanwise direction within a certain range correlate negatively to each other.
This negative correlation is preserved under unstable stratification in the present cases, as shown in figure 4(b). Though
not shown here, the negative peak of correlation locates atry /δ ≈ 0.4, irrespectively of the value of δ /L and the height ofz for the unstable cases. The negative correlation is therefore
associated to δ -scaled large scale structures.
The effect of unstable stratification is also observed in
Cvv . As seen in figure 5(a), the positive correlation becomesstronger, and correlation length in the streamwise direction iselongated as −δ /L increases. For the spanwise distance, as
shown in figure 5(b), only the most unstable case U3 significantly deviates from the others, and gives a longer correlation length. Furthermore, Cvv (0, ry ) at ry /z = 5 is found to beslightly negative for the unstable cases. The both correlation
lengths in the x and y directions of v become longer by thebuoyancy effect. These observation also evidence the emergence of coherent roll-like structures.

REFERENCES
Blass, A., Zhu, X., Verzicco, R., Lohse, D. & Stevens, R. J.
A. M. 2020 Flow organization and heat transfer in turbulent
wall sheared thermal convection. J. Fluid Mech. 897, A22.
Etling, D. & Brown, R. A. 1993 Roll vortices in the planetary
boundary layer: a review. Boundary-Layer Meteorol. 65,
215–248.
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013
On the logarithmic region in wall turbulence. J. Fluid Mech.
716, R3–1–11.
Mizuno, Y., Yagi, T. & Mori, K. 2022 Momentum flux in turbulent boundary layers with weakly unstable stratification.
J. Phys. Soc. Jpn 91, 054402.
Monin, A. S. & Obukhov, A. M. 1954 Basic laws of turbu-

CONCLUSIONS
It is found that the vertical profile of the variance and also
spectrum of the streamwise velocity component are not modified by the stratification up to −δ /L ≈ 1. This indicates thatthe mechanism of energy transfer of the streamwise velocity
component in a neutral case is retained in that range of −δ /L.
On the other hand, they exhibit a qualitative change beyond

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

1

1
(a)

0.6
0.4
0.2
0
-0.2
0

(b)

0.8
Cuu(0, ry )Cuu(rx, 0)

0.8

0.6
0.4
0.2
0

1

2

3

4

5

6

-0.2
0

7

1

2

rx/z
Figure 4.

5

6

1
(a)

0.8

(b)

0.8

0.6

Cvv (0, ry )Cvv (rx, 0)

4

Correlation function (a) Cuu (rx , 0) and (b) Cuu (0, ry ) at z/δ ≈ 0.15. Symbols are as in Table 1.

1

0.4
0.2
0
-0.2
0

3
ry /z

0.6
0.4
0.2
0

1

2

3

4

5

6

-0.2
0

7

1

rx/z
Figure 5.

2

3
ry /z

4

5

6

Same as figure 4, but for Cvv .

lent mixing in the atmosphere near the ground. Trudy geofiz.
inst. AN SSSR 24, 163–187.
Pirozzoli, S., Bernardini, M., Verzicco, R. & Orlandi, P. 2017
Mixed convection in turbulent channels with unstable stratification. J. Fluid Mech. 821, 482–516.
Sillero, J. A., Jiménez, J. & Moser, R. D. 2013 One-point
statistics for turbulent wall-bounded flows at Reynolds
numbers up to δ + ≈ 2000. Phys. Fluids 25, 105102.

Sillero, J. A., Jiménez, J. & Moser, R. D. 2014 Two-point
statistics for turbulent boundary layers and channels at
Reynolds numbers up to δ + ≈ 2000. Phys. Fluids 26,
105109.
Townsend, A. 1976 The Structure of Turbulent Shear Flow,
2nd edn. Cambridge: Cambridge University Press.
Wyngaard, J. C. 2010 Turbulence in the Atmosphere. Cambridge: Cambridge University Press.

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