Experimental observation on Experimental observation on ppsurface undulations over surface undulations over 

erodible bederodible bed

Annual Workshop of  Advanced Modeling on Shear Shallow Flows Annual Workshop of  Advanced Modeling on Shear Shallow Flows 

on Curved Topography on Curved Topography AMoSSAMoSS

(INRIA(INRIA‐‐MOST 2014MOST 2014‐‐2016) Associated Team Work2016) Associated Team Work

8th July 2015, in Antibes/Nice, France

Yih‐Chin TAI1, Chin‐Kai Cheng1 and Chih‐Yu Kuo21Dept. Hydraulic & Ocean Eng., National Cheng Kung University, Tainan 701, Taiwan

2Research Center for Applied Sciences, Academia Sinica, 115 Taipei, Taiwan

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

Water‐sand mixtureSand material : Ottawa sand (C‐778, diameter: 06 – 0.8 mm)Sand material  :  Ottawa sand (C 778, diameter: 06  0.8 mm)Plexi‐glass channel  :  1.6 m long, 1 cm wideChannel slope  :  3 – 5 degreeWater flux  :  0.00898 kg/sSand supply  :  0.3 kg/s

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

30cm 5cm 45cm20cm

Experimental observation• Steady state• high‐speed camera (fps = 200)Ch l l 0 d• Channel slope: 0 degree

• Bed slope: ca. 2.5 degreeTh f d l ti i hibit d b th St f b ll• The surface undulation is exhibited by the Styrofoam ball

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

• Channel slope: 3 degree,    Bed slope:  ca. 1 degree,   fps = 1,000

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

• Channel slope: 3 degree,    Bed slope:  ca. 1 degree,   fps = 1,000Trough 1 Trough 2 Trough 3Crest 1 Crest 2

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

FrouheFrouhe Number (only approximation)Number (only approximation)• Piecewise linear distribution u uPiecewise linear distribution wu

2 2( )d ,s

b

z

zu z z u h

shh 21 2 3 1 0.5 ,h h h sh h

2Fr cos 3 1u gh u

• Channel slope: 3 degree

s g

p g

Trough 01 0 4874 0 0111 0 0049 1 1612 1 0346

(m / s)u (m)h (m)sh FrTrough 01 0.4874 0.0111 0.0049 1.1612 1.0346

Crest 01 0.5091 0.0125 0.0052 1.1517 1.0387

Trough 02 0.4729 0.0106 0.0057 1.1977 0.9721

Crest 02 0.4986 0.0127 0.0050 1.1451 1.0352

Trough 03 0.4977 0.0106 0.0047 1.1641 1.0473

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

• Channel slope: 5 degree,    fps = 1,000

Crest no.1 Trough no.1 Crest no.2 Trough no.2

Crest 01 0.6830 0.0101 0.0040 1.1427 1.2504

(m / s)u (m)h (m)sh Fr

Trough 01 0.6592 0.0091 0.0040 1.1588 1.211

Crest 02 0.6513 0.0101 0.0040 1.1427 1.2303

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

Trough 02 0.6644 0.0096 0.0040 1.1507 1.2268

• Velocity and pressure measurements in undular hydraulic jump after Fawer (1937)after Fawer (1937) 

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

Conventional method: Boussinesq‐type modele g Serre (1953) Peregrine (1966) etc

hUth

0

e.g. Serre (1953), Peregrine (1966), etc.

( ) dz

w z u z

bxxxtxhUUUUghhU

xthU

xt

321 3

222

( ) dbz

b

w z u zx

U z zx

Saint‐Venant term        non‐hydrostatic term

x

Shear Flow model: Richard and Gavrilyuk (2012, 2013)h h

hghhUhU

hUxt

h

322 1

0

0

Uhh

Uhxt

h

bhghhUxt

2 0

Uh

xt

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN

Open questions forOpen questions forOpen questions for Open questions for discussiondiscussion

• Is the observation a three-dimensionalphenomena?

Looking for optimal model• Looking for optimal model

• The wavy formation at the bottomy

• The impacts of the sediment-water layer.

• Rigid bed and erodible bottom.

Dr.-Ing. Y. C. Tai, Dept. Hydraulic and Ocean Engineering, NCKU, TAIWAN