# Turbulent Coherent Structures in a Thermally Stable Boundary Layer

**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.

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**ABSTRACT:**The critical Richardson number, Ric, is used in studies of stably stratified turbulence as a measure of flow laminarization. The accepted range of Ric is between 0.2 and 1. A growing body of experimental and observational data indicates, however, that turbulence survives for Ri ≫ 1. This result is supported by a new spectral theory of turbulence that accounts for strong anisotropy and waves. The anisotropization results in the enhanced horizontal mixing of both momentum and scalar. Internal wave contribution preserves vertical momentum mixing above its molecular level. In the absence of laminarization, Ric becomes devoid of its conventional meaning. Copyright © 2007 Royal Meteorological SocietyAtmospheric Science Letters 07/2007; 8(3):65 - 69. · 1.88 Impact Factor - Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences 01/1920; 97:354-373. · 2.00 Impact Factor
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**ABSTRACT:**Observations made in a well-developed, thermally stratified, horizontal, flat- plate boundary layer are used to study the effects of buoyancy on the mean flow and turbulence structure. These are represented in a similarity framework obtained from the concept of local equilibrium in a fully developed turbulent flow. Mean velocity and temperature profiles in both the inner and outer layers are strongly dependent on the thermal stratification, the former suggesting an increase in the thickness of the viscous sublayer with increasing stability. The coefficients of skin friction and heat transfer, on the other hand, decrease with increasing stability.Journal of Fluid Mechanics 03/1975; 68(02):321 - 343. · 2.29 Impact Factor

Page 1

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 < Riδ< 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, U is the mean velocity, z is the

wall-normal distance and g is 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

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Figure 1.

plate

PIV setup to measure turbulent statistics of a thermally stable boundary layer developing on the underside of a heated

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,

Riδ=gδΘ∞

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

firstanalysisofturbulentstratifiedflowsbyRichardson(1920)

predictedthatforRi>1noturbulencewouldsurvive. 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=

gθw

uw∂U

Θ

∂z

(3)

Here, θw is the turbulent heat flux, Θ is 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) (Riδ< 0.1)

2

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Experimental Case NV4T1V4T2V4 T3V4T4V4 T5V4T6V4 T7V4

∆Θs

0 19.7 42.860.476.3 100.0115.5128.9

Reθ

1032986 1048990 901907 813773

Riδ

uτ/U∞

0 0.0087 0.035 0.0480.0600.077 0.088 0.097

0.048-------

Table 1.Flow properties with varying wall temperature but constant velocity of U∞= 1.44m/s

Experimental CaseT3V1T3V2T3V3 T3V4 T3V5T3V6T3V7T3V8 T3V9

U∞(m/s)

0.96 1.121.281.43 1.61 1.782.08 2.422.63

Reθ

696774.8841990 1169 13701533 1803 2040

Riδ

0.11 0.079 0.0610.0480.038 0.0310.0230.0170.014

Table 2.Flow properties with varying velocity but constant ∆Θs= 60

and also Ogawa et al. (1985) (Riδ< 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) (Riδ< 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, Riδ= 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 ≤ Riδ≤ 0.2 and 600 ≤ Reθ≤ 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 300µs. 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-wallseedingwasfoundtobeinsufficientduetothe

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

Page 4

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 Riδ= 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 Reθagainst Riδfor 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 ≤ Reθ≤ 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.

that the remaining turbulence is uncorrelated noise and the

It is possible

600 80010001200 1400

Reθ

1600 1800 20002200

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Riδ

Weakly Stable

Strongly Stable

Figure 3.

studied here.

Reynolds and Richardson numbers for all cases

Figure 4.

0.048). Symbols as in Figure .

Structure of boundary layer for T3V4 (Riδ=

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-

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Page 5

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 Riδ.

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

ismostlikelyconnectedwithfluctuationsinthelocalRichard-

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 Riδ= 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.

REFERENCES

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00.20.4 0.60.81

0

0.1

0.2

0.3

0.4

0.5

0.6

U/U∞

Z/δ

T3V2

T3V4

T3V6

T3V7

T3V8

T3V9

0 0.20.40.6 0.81

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)

00.020.040.06

u′/U∞

0.08 0.10.12

0

0.1

0.2

0.3

0.4

0.5

0.6

Z/δ

T3V2

T3V4

T3V6

T3V7

T3V8

T3V9

00.020.040.06

u′/U∞

0.08 0.10.12

0

0.1

0.2

0.3

0.4

0.5

0.6

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 0.20.40.6 0.8

−u′ v′/U∞

1 1.21.41.61.8

−3

x 10

0

0.1

0.2

0.3

0.4

0.5

0.6

2

Z/δ

T3V2

T3V4

T3V6

T3V7

T3V8

T3V9

00.20.40.60.8

−u′ v′/U∞

11.2 1.41.61.8

−3

x 10

0

0.1

0.2

0.3

0.4

0.5

0.6

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

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(a) Neutrally stable boundary layer

(b) Weakly stable (Riδ= 0.014)

(c) Strongly stable (Riδ= 0.08)

Figure 5.

vorticity contours, swirl strength and a Galilean transform of the velocity field.

Turbulent structures within stratified and neutrally stable boundary layers. These are visualized using a combination of

7

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