1. Sheared stably stratified turbulence and
BEN-GURION UNIVERSITY OF THE NEGEV FACULTY OF ENGINEERING SCIENCES DEPARTMENT OF MECHANICAL ENGINEERING
Sheared stably stratified turbulence and
large-scale waves in a lid driven cavity
N. Cohen, A. Eidelman, T. Elperin, N. Kleeorin, and I. Rogachevskiie
Hello my name is Nimrod and I would like to talk on Sheared stably stratified turbulence and large-scale waves in a lid driven cavity
First Thermal and Fluids Engineering Summer Conference (TFESC) New-York, July 9-12, 2015

2. Contents Introduction Experimental set-up Experimental results
Review
Motivation
Experimental set-up
Scheme of the experimental set-up
Velocity fields
Measurement technique
Temperature fields
Experimental results
Summary and conclusions
I will start with short motivation to this work.
Then I will present the experimental set up.
Afterwards I will present the experimental results.
In the end I will summary's.
Waves

3. Lid Driven Cavity (LDC) flows
Flow and heat transfer analysis in lid-driven cavities (LDC) is one of the most widely studied problems in thermo-fluids.
LDC configuration is encountered in many practical engineering and industrial applications and serves as a benchmark problem for numerical simulations.
Numerous investigations have been conducted in the past considering various combinations
R. Viskanta
R. Iwatsu
J. Miles
However there are only a few experimental studies on stably-stratified flows in lid-driven cavity:
J. R. Koseff
LDC is one of the most widely studied problems in thermo-fluids.
LDC configuration is encountered in many practical engineering and industrial applications and serves as a benchmark problem for numerical simulations.
Numerous investigations have been conducted in the past considering various combinations .
However there are only a few experimental studies on stably-stratified flows in lid-driven cavity.

4. Sheared turbulence flow in a stably stratified temperature in a lid-driven cavity
A non-zero vertical mean temperature gradient is imposed such that the shear driven and buoyancy effects are of comparable magnitude (mixed convection regime).
We investigated experimentally effects of Richardson number on the mean and turbulent flows, momentum and heat transfer.
The obtained result my be useful in atmospheric applications. g
Such configuration can be idealized by the simple rectangular geometry with regular boundary conditions yielding a well-posed problem.
We imposed non-zero vertical temperature gradient such that the shear driven and buoyancy effects are of comparable magnitude (mixed convection regime).
Object of this research is to investigated experimentally effects of Richardson number on the mean and turbulent flows, momentum and heat transfer
The obtained result my be useful in atmospheric applications

5. Scheme of the experimental set-up
This experimental set-up allows us to create a sheared turbulence flow in a stably stratified temperature.
The top wall of the cavity was heated and moves in minus Y axis direction to generates a shear flow in the cavity
cavity filled with air
This experimental set-up allows us to create a sheared turbulence flow in a stably stratified temperature.
In the center is the cavity filled with air.
The bottom wall of the cavity was cooled .
The top wall of the cavity was heated and moves in minus Y axis direction to generates a shear flow in the cavity.
The bottom wall of the cavity was cooled

6. The experimental facility
Rectangular cavity :
Constant lid velocity (top wall of the cavity) :
Magnitudes of the imposed temperature difference:
Bulk Richardson number :
Here you can see the experimental facilitys in our lab.
We use rectangular cavity with those dimension.
The lid velocity was constant and set to 1.13 m/s.
In each experiment we set a temperature different dT between the wall .
Bulk Richardson number is varied
Here β is thermal expansion coefficient

7. Velocity fields measurement
The turbulent velocity field have been measured using a digital Particle Image Velocimetry (PIV) system with LaVision Flow Master III.
Flow velocity field was measured in:
Side cross sections
Frontal central cross section
Flow velocity field was measured in different side cross sections (x-z plane) along the cavity.
Flow velocity field was measured in a frontal central cross section (y-z plane) of the cavity.
We used PIV system to measured velocity field in front and side cross section.

8. Temperature field measurements
Temperature field was measured in a frontal central cross section using temperature probe equipped with E-type thermocouples.
The exact position of each thermocouple was measured using images captured with the optical system employed in PIV measurements
We used probe equipped with E-type thermocouples to measured temperature field in front cross section.

9. z y Instantaneous images Instantaneous velocity field
Instantaneous images z
Instantaneous velocity field y
Instantaneous velocity fluctuations field
In each experiment we record a serial of 500 images.
In this move Ri set to 0.22 and we are looking on FV

10. Mean velocity field in a frontal central cross section
Experimental results
Mean velocity field in a frontal central cross section
(isothermal case)
Ri=0 (min)
Ri=0.035
Ri=0.064
Ri=0.093
Ri=0.121
Ri=0.294 (max)
Ri=0.244
Ri=0.220
Ri=0.186
Ri=0.148
In this slide we show the mean velocity field in FV for each Ri number were we used ensemble averaging; notice that Ri increase clockwise .
The stable stratification suppresses vertical motions, and, therefore, the impact of the sliding top wall penetrates to smaller distance into the fluid.
When Ri number is large, the flow in the middle and lower parts of the cavity interior is weak, and much of the fluid here remains almost stagnant.
The stable stratification suppresses vertical motions.
When Ri number is large, the flow in the middle and lower parts is weak, and much of the fluid remains almost stagnant.

11. Experimental results
Mean velocity field in a frontal central cross section Vs. Richardson number
This movie serve to identify the major qualitative changes in the global flow patterns as Ri encompasses a wide range.
The stable stratification suppresses vertical motions, and, therefore, the impact of the sliding top wall penetrates to smaller distance into the fluid. As seen. When Ri is large, the flow in the middle and lower parts of the cavity interior is weak, and much of the fluid here remains almost
stagnant. This movie exemplfies this trend

12. Experimental results cm K Using the energy budget estimate we obtain:
Maximum vertical size of large-scale circulation cm
25 K
Inspection shows that the maximum vertical size
Lz of large-scale circulation is nearly constant when T < 25 K. On the other hand, when
T > 25 K, the maximum vertical size Lz decreases with dT as Lz ∝ 1/T.
16 K
33 K K
Using the energy budget estimate we obtain:

13. Experimental results
FV, Ri=0
FV, Ri=0.148
FV, Ri=0.294
Here we show sketches of the mean flow pattern in a side and frontal cross-sections of the cavity for 3 different Ri number.
SV, Ri=0
SV, Ri=0.148
SV, Ri=0.294

14. Temperature profiles V V V
Spatial vertical profiles of mean temperature field in the
central cross-section for different y V V V
Heated top wall
Heated top wall
Heated top wall
In the main vortex the fluid is well mixed and temperature variations are very small.
Outside the main vortex you can notice significant vertical temperature gradient .
Cooled bottom wall
Cooled bottom wall
Cooled bottom wall

15. Internal gravity waves
Internal gravity waves in stably stratified flows satisfied the follow dispersion relation.
Region of the cavity with weak turbulence is the regions outside the large-scale vortex (lower part of the cavity) which in it
We define the following functions :
Normalized one-point non-instantaneous correlation function of the large-scale temperature field :
Spatial vertical profiles of mean temperature field T(z) in the central cross-section for different y at the temperature difference K
y=2.5 cm (triangles)
y=10 cm (squares)
y=14 cm (diamonds)
y=22 cm (stars)
Internal gravity waves in stably stratified flows satisfied the follow dispersion relation .
Notice that the vertical temperature gradient is different form 0 in lower part of the cavity which is the region outside the large-scale vortex .
We define the following functions : cm

16. Internal gravity waves
Normalized one-point non-instantaneous correlation function of the large-scale temperature field determined for different z versus the time at the temperature difference 54 K (y=10 cm).
Z=2.5 cm (diamonds)
Z=7.4 cm (crosses)
Z=15.9 cm (squares)
Z=5.1 cm (six-pointed stars)
Z=11.9 cm (snowflakes)
Z=18.9 cm (circles)
[s]
[s]
Here we plot the correlation function
has a form of the function :
Dashed line fitting with :
which corresponds to the period of the wave

17. Internal gravity waves
Let us define the following functions :
The normalized one-point non-instantaneous correlation function of the vertical large-scale velocity field.
has a form of :
Dashed line fitting with :
and which corresponds to the period of the wave
Normalized one-point non-instantaneous correlation function of the vertical large-scale velocity field versus the time at the temperature difference 54 K (y=10 cm, z=2.5 cm).
We define the following functions :
[s]

18. Internal gravity waves
For time scales these correlation functions are different.
This implies that fore these time-scales the wave spectra of the large-scale velocity and temperature fields are different.
Comparison of and versus the time
Comparison of these correlation functions
[s]

19. Internal gravity waves
is Fourier transform of the normalized one-point non-instantaneous correlation function
Fourier transform of the function has the following form:
Single peak for etch variable.
The frequency ratio of these peak :
The spectral functions
The spectral function
[1/s]

20. Summary and conclusions
Richardson number is a measure of the relative strength of buoyancy-driven natural convection and lid-driven forced convection.
When Ri >>1, buoyancy forces are clearly dominant; When Ri <<1, the shear effect dominates; When Ri is of the order of unity, the free (buoyant) and forced convection effects are of comparable magnitude.
Geometrical properties of the large-scale vortex (e.g., its size and form) are controlled by the buoyancy.
The observed velocity fluctuations are produced by the shear of the large-scale vortex.
At larger stratification obtained in our experiments (Ri=0.294):
Strong turbulence region is located at the upper left part of the cavity where the large-scale vortex exists and the temperature field I is fairly homogeneous
At the upper left part of the cavity the Brunt–Väisälä frequency is small and increases in the direction outside the large-scale vortex.
By analyzing the correlation functions of temperature and velocity fields we found internal gravity waves in the system.
This form of the correlation functions indicates the presence of the large-scale waves.
The large-scale internal gravity waves are observed in the regions outside the large-scale vortex.
The behavior of correlation functions is the same at the time interval of 10 s.
The observed large-scale waves are nonlinear because the frequency of the waves are different

21. Questions ?

22. Experimental results
Characteristic horizontal mean and turbulent velocities
Integral scales of turbulence in horizontal and vertical directions
cm/s mm
25 K
25 K K K
We estimate the turbulent kinetic energy using the budget equation:
The turbulent velocity increases with the increase of temperature difference.
We estimate the turbulent kinetic energy using the budget equation for this quantity.
The turbulent velocity increases with the increase of the stratification.

23. Internal gravity waves
Spatial vertical profiles of turbulent temperature fluctuations and of the function
for different z at the temperature difference 33 K (y=6 cm).
Intensities of turbulent temperature fluctuations are of the same order as the temperature fluctuations
in the large-scale internal gravity waves.
Turbulent fluctuations are larger in:
Lower part of the cavity where the mean temperature gradient is maximum.
Upper part of the cavity where the shear caused by the large-scale circulation is maximum. cm
Intensities of turbulent temperature fluctuations are of the same order as the temperature fluctuations
in the large-scale internal gravity waves.

24. Summary and conclusions
Richardson number is a measure of the relative strength of buoyancy-driven natural convection and lid-driven forced convection.
When Ri >>1, buoyancy forces are clearly dominant; When Ri <<1, the shear effect dominates; When Ri is of the order of unity, the free (buoyant) and forced convection effects are of comparable magnitude.
Geometrical properties of the large-scale vortex (e.g., its size and form) are controlled by the buoyancy.
The observed velocity fluctuations are produced by the shear of the large-scale vortex.
The level of small-scale turbulence inside the vortex are controlled by the buoyancy.
At larger stratification obtained in our experiments (Ri=0.294):
Strong turbulence region is located at the upper left part of the cavity where the large-scale vortex exists and the temperature field I is fairly homogeneous
At the upper left part of the cavity the Brunt–Väisälä frequency is small and increases in the direction outside the large-scale vortex.
By analyzing the correlation functions of temperature and velocity fields we found internal gravity waves in the system.
This form of the correlation functions indicates the presence of the large-scale waves.
The measured intensity of the waves is of the order of the level of the temperature turbulent fluctuations.
The large-scale internal gravity waves are observed in the regions outside the large-scale vortex.
The behavior of correlation functions is the same at the time interval of 10 s.
The observed large-scale waves are nonlinear because the frequency of the waves determined from the temperature field measurements is two times smaller than that obtained from the velocity field measurements.