On Reynolds Stresses over Wind Waves

 

On Reynolds Stresses over Wind Waves

 

Tel-Aviv University

School of MechanicalEngineering

Supported by Israel Science Foundation

Lev Shemer and Andrey ZavadskyLev Shemer and Andrey Zavadsky

TAU Wind-Wave Flume

Test Section Dimensions:

5 m Long, 0.4 m Wide, 0. 58 m High

Water Depth 0.2 m Max Wind Speed: > 15 m/s

View of the sensors

Pitot tube

X-hot film

Static pressure sensor

Capacitance wave gauges

Max. crest height detector

View within the test section in operating conditions

Experimental conditions7 Fetches (x) Each 40 cm; 100 cm ≤ x ≤ 340 cm

4 Wind VelocitiesU10= 16.5 m/s; 18.6 m/s; 21.0 m/s; 25.4 m/s

At each fetch, 40 vertical locations

5 mm ≤ z<≈ 150 mm above the highest crest

Simultaneously Recorded Parameters

Temperature is maintained constant for thermo-anemometry

Surface Elevation η, Static pressure p, instantaneous horizontal u and vertical w air velocities

Continuous SamplingDuration: 5 – 10 min @ sampling rate of 120 Hz or 1200 Hz

Automatic sensor calibration and data recording at each fetch

Measurement session for 4 wind speeds and all elevations lasts overnight (about 16 hours)

Vertical Air Velocity Profiles

101 1026.5

7

7.5

8

8.5

9

9.5

Height above Mean Water Level , mm

Ua,

m/s

U10

= 18.6 m/s

x=100 cmx=140 cmx=180 cmx=220 cmx=260 cmx=300 cmx=340 cm

101 1028.5

9

9.5

10

10.5

11

11.5

12

Height above Mean Water Level, mm

Ua,

m/s

U10

=25.4 m/s

x=100 cmx=140 cmx=180 cmx=220 cmx=260 cmx=300 cmx=340 cm

Vertical Distribution of –u’w’

0 20 40 60 80 100 120 140 160-0.05

0

0.05

0.1

0.15

0.2

Height above Mean Water Level, mm

-u'w

' m2/s

2

U10

= 16.5 m/s

x=100 cmx=140 cmx=180 cmx=220 cmx=260 cmx=300 cmx=340 cm

0 20 40 60 80 100 120 140 160-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-u'w

', m

2/s

2

Height above Mean Water Level, mm

U10

= 25.4 m/s x=100 cmx=140 cmx=180 cmx=220 cmx=260 cmx=300 cmx=340 cm

Friction Velocity u* (m/s)

100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1

Fetch, cm

u*,

m/s

Log Profile U10

=16.5 m/s

U10

=18.6 m/s

U10

=21.0 m/s

U10

=25.4 m/s

Reynolds Stress U10

=16.5 m/s

U10

=18.6 m/s

U10

=21.0 m/s

U10

=25.4 m/s

The values of the friction velocity u*

determined from fitting the log velocity profile (κ=0.4) and from the Reynolds stress measurements agree within 10-20%

Full symbols – from log velocity profile; Empty symbols – from Reynolds stresses

Air velocity fluctuations

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

Z, mm

(w'2

)1/2

m/s

x=100cmx=140cmx=180cmx=220cmx=260cmx=300cmx=340cm

0 50 100 1500

0.2

0.4

0.6

0.8

1

Z, mm

(u'2

)1/2

m/s

x=100cmx=140cmx=180cmx=220cmx=260cmx=300cmx=340cm

u’ w’U10=16.5 m/s

0 50 100 1500

0.2

0.4

0.6

0.8

1

Z, mm

(w'2

)1/2

m/s

x=100cmx=140cmx=180cmx=220cmx=260cmx=300cmx=340cm

0 50 100 1500

0.5

1

1.5

Z, mm

(u

'2)1

/2 m

/s

x=100cmx=140cmx=180cmx=220cmx=260cmx=300cmx=340cm

U10=25.4 m/s

Records of velocity components u and w and pressure fluctuations p in air compared to the

surface elevation η; U10=25.4 m/s

100 100.5 101 101.5 102 102.5 103 103.5 104 104.5 105-20

-10

0

10

20

30

40

t, sec

u, m/sw, m/s, mm Pressure, Pa

Simultaneous measurements at x=300cm, 5 mm above the highest crest

Cross-Spectral AnalysisCross-Spectral Analysis

Measured records of 2 signals:

f(t) = f(ti) and g(t) = g(ti); ti = (i-1)Δt; i=1,…N

2

2

1( ) ( ) ( )

T

fgT

R t f g t dT

Cross-correlation function

and cross-spectrum

Magnitude Squared Coherence (MSC)

2

2

1( ) ( )

Ti t

fg fgT

S R t e dtT

22 ( )

( )( ) ( )

fgfg

f g

S

S S

Coherence between turbulent velocity components u’, w’ and surface elevation

ηU10=16.5 m/s U10=25.4 m/s

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

frequency, [Hz]

MS

C (

-u)

z=5mmz=6mmz=7mm

η - u

η - w

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

frequency, [Hz]

MS

C (

-w)

z=5mmz=6mmz=7mm

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

frequency, [Hz]

MS

C (

-u)

z=5mmz=6mmz=7mm

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency, [Hz]

MS

C (

-w)

z=5mmz=6mmz=7mm

Phase relations between surface elevation and air velocity fluctuations

240 260 280 300 320 340 360-60

-40

-20

0

20

40

60

80

fetch, [cm]

, [

de

g]

w

; U10

=16.5 m/s

w

; U10

=18.6 m/s

w

; U10

=21.0 m/s

w

; U10

=25.4 m/s

u

; U10

=16.5 m/s

u

; U10

=18.6 m/s

u

; U10

=21.0m/s

u

; U10

=25.4 m/s

“Coherent” wave-induced velocity fluctuations do not contribute to Reynolds shear stress

Conclusions Detailed experimental results on the structure of

turbulent air flow over waves are presented, including spatial distributions of normal and shear Reynolds stresses and phase relations between waves and air flow

To enable those measurements, a fully automated procedure was developed to control the experimental conditions and to record data simultaneously from numerous sensors

These results can serve a basis for validation of theoretical models dealing with wind-wave interactions

Extensive further experiments, also involving mechanically generated waves are now in progress

ThankThank you !you !