STABLY STRATIFIED SHEAR-PRODUCED TURBULENCE AND LARGE-SCALE-WAVES IN A LID DRIVEN CAVITY

BEN-GURION UNIVERSITY OF THE NEGEV FACULTY OF ENGINEERING SCIENCES DEPARTMENT OF MECHANICAL ENGINEERING

N. Cohen, A. Eidelman, T. Elperin, N. Kleeorin, and I. Rogachevskiie

15th European Turbulence Conference August 25-28th, 2015, Delft, The Netherlands.

Scheme of the experimental set-up

Introduction

Waves

Review

Motivation

Experimental set-up

Contents

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

Lid Driven Cavity (LDC) flows

Sheared turbulence flow in a stably stratified temperaturein a lid-driven cavity

g

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.

Scheme of the experimental set-up

This experimental set-up allows us to create a sheared turbulence flow in a stably stratified temperature.

cavity filled with air

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

The experimental facility

• Rectangular cavity :

• Constant lid velocity (top wall of the cavity) : .

• Magnitudes of the imposed temperature difference:

Bulk Richardson number : 294.00 RiHere β is thermal expansion coefficient

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:

Flow velocity field was measured in a frontal

central cross section (y-z plane) of the cavity.

Flow velocity field was measured in different side

cross sections (x-z plane) along the cavity.

mmy 140,100,80,60,40,25

Velocity fields measurement

Frontal central cross sectionSide cross sections

• 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

Temperature field measurements

Instantaneous images

Instantaneous velocity field y

z

Instantaneous velocity fluctuations field

Experimental results

Ri=0 (min) Ri=0.035 Ri=0.064 Ri=0.093 Ri=0.121

Ri=0.148Ri=0.186Ri=0.220Ri=0.244Ri=0.294 (max)

•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.

Mean velocity field in a frontal central cross section

(isothermal case)

Experimental results

Mean velocity field in a frontal central cross section Vs. Richardson number

Experimental results

Maximum vertical size of large-scale circulation

16 K 33 K25 K

K

cm

Using the energy budget estimate we obtain:

Experimental results

FV, Ri=0

SV, Ri=0 SV, Ri=0.148

FV, Ri=0.148 FV, Ri=0.294

SV, Ri=0.294

Temperature profiles

Spatial vertical profiles of mean temperature field in the central cross-section for different y

Heated top wall

Cooled bottom wall

Heated top wall

Cooled bottom wall

Heated top wall

Cooled bottom wall

)26(148.0 KTRi )33(187.0 KTRi )5.39(220.0 KTRi V V V

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 :

Internal gravity waves

Spatial vertical profiles of mean temperature field T(z) in the central cross-section for different y at the temperature difference

cm

K

y=2.5 cm (triangles) y=10 cm (squares) y=14 cm (diamonds) y=22 cm (stars)

Internal gravity waves

has a form of the function :

Dashed line fitting with :

which corresponds to the period of the wave

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).

[s] [s]

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)

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

Internal gravity waves

[s]

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).

• 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.

Internal gravity waves

[s]

Comparison of and versus the time

Internal gravity waves

[1/s]

The spectral functions

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 :

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

Questions

?

Experimental results

Characteristic horizontal mean and turbulent velocities

K

cm/s

Integral scales of turbulence in horizontal and vertical directions

K

25 K 25 K

We estimate the turbulent kinetic energy using the budget equation:

The turbulent velocity increases with the increase of temperature difference.

 

mm

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.

Internal gravity waves

cm

Spatial vertical profiles of turbulent temperature fluctuations and of the function

for different z at the temperature difference 33 K (y=6 cm).

Change of the mean kinetic energy :

Work of the buoyancy force :

Experimental results