Water waves over a muddy seabed

Chiang C. Mei, MIT

WithZhenhua Huang, NTU, Singapore M.Krotov, MITErell-Isis Garnier, MITTFunded by US Office of Naval Research

April,May, 2013 ROC, Toronro 1

Allepey, S. India (Mathew, Baba & Kurian, 1995)

April,May, 2013 ROC, Toronro 2

India, Mathew,

Baba, Kurian

JCR 1995

April,May, 2013 ROC, Toronro 3

5 6 7 8

April,May, 2013 ROC, Toronro 4

1989

Alleppey, S. India (Mathew et al 1995)

April,May, 2013 ROC, Toronro 5

Past works

• Field survey: – Surinum: Wells & Colemen 1981

– India: Mathew, Baba & Kurian, 1995

• Math models:

• Newtonian mud: Dalrymple & Liu 刘立方 , Liu & Chan

– Bingham plastic: Mei and Liu 刘格非 , Coussot

– Simple Kelvin-Voigt viscoelastic: McPherson, Jiang & Mehta, Ng 吴朝 安

– Plasto-viscoelastic: Shibayama

April,May, 2013 ROC, Toronro 6

April,May, 2013 ROC, Toronro 7

Lianyun gang

Hangzhou bay

Field mud samples

连云港

杭州湾

Fluid-mud rheologyNot Newtonian, Not Bingham plastic, but

Viscoelastic

April,May, 2013 ROC, Toronro 8

April,May, 2013 9ROC, Toronro

April,May, 2013 ROC, Toronro 10

Part I: Horizontal seabed, intermediate wave length

(Mei, Krotov & Huang, 2010, JFM)

April,May, 2013 ROC, Toronro 11

Shallow mud layer

April,May, 2013 ROC, Toronro 12

Scales

April,May, 2013 ROC, Toronro 13

Water layer

April,May, 2013 ROC, Toronro 14

Water at Leading orderHomogeneous BVP on the fast scale

April,May, 2013 ROC, Toronro 15

Leading order mud motion

April,May, 2013 ROC, Toronro 16

Stokes boundary layer

April,May, 2013 17ROC, Toronro

April,May, 2013 ROC, Toronro 18

Solvability of order 1

April,May, 2013 19ROC, Toronro

Interface displacement

Hangzhou Bay for

d/h=0.03--0.2

April,May, 2013 ROC, Toronro 20

Interface displacement

Lianyun Harborfor

d/h=0.03—0.2

April,May, 2013 ROC, Toronro 21

April,May, 2013 22ROC, Toronro

Damping coefficient

Part II

Shallow water long waves on a sloping beach

Garnier, Huang & Mei, J. Fluid Mech. 2013,(718), 371-397.

April,May, 2013 ROC, Toronro 23

April,May, 2013 ROC, Toronro 24

Boussinesq approximation

Shallow water, very shallow mud:

April,May, 2013 ROC, Toronro 25

Fluid mud from Hangzhou Bay and Lianyun Harbor (Huang & Huhe

1992)

April,May, 2013 26

Lianyun Harbor

Hangzhou Bay

Lianyun Harbor

Hangzhou Bay

ROC, Toronro

Hangzhou Bay Hangzhou Bay

Lianyun HarborLianyun Harbor

Water layer:Long wave equations

(dimensionless and Boussinesq)

April,May, 2013 ROC, Toronro 27

Crudest approximation

April,May, 2013 28ROC, Toronro

Harmonic generation in shallow water

April,May, 2013 ROC, Toronro 29

frequency

Wave number

April,May, 2013 ROC, Toronro 30

(a) A = 5 cm: H = 30 cm, T = 1:90 s,L = 3:23 m (Ur = 0:45);

(b) A = 5 cm: H = 20 cm, T = 2:75 s, L= 3:86 m (Ur = 2:35)

Barbara Karakiewicz, 1972

Periodic surface waves

April,May, 2013 ROC, Toronro 31

Mud motion at leading order

April,May, 2013 ROC, Toronro 32

Mud Harmonics

April,May, 2013 ROC, Toronro 33

Wave harmonics

April,May, 2013 ROC, Toronro 34

April,May, 2013 ROC, Toronro 35

Hangzhou Bay Lianyun Harbor

d=0.5 m

d=0.75 m

d=1.0 m

Surface wave harmonics

April,May, 2013 ROC, Toronro 36

Hangzhou Bay Lianyun Harbor

Interface harmonics

April,May, 2013 ROC, Toronro 37

Near the shoreline

April,May, 2013 ROC, Toronro 38

Steady displacement (cf: Acoustic streaming)

April,May, 2013 ROC, Toronro 39

Mean mud displacement

April,May, 2013 40ROC, Toronro

Profile of mud surface displacement

April,May, 2013 ROC, Toronro 41

angzhou Bay

Ianyun Harbor

Work needed

• Short-range:– Measurement of fluid mud rheology with a

dynamic rheometer (RMS 605?)

– Laboratory experiments on a sloping beach

• Fundamental:– Transition between consolidated and fluidized

mud

– Transition between fluid mud and clear water (lutocline)

April,May, 2013 ROC, Toronro 42

April,May, 2013 ROC, Toronro 43

Thank youMerci， GraciasVielen Dankありがとうございます。

谢， 감사합니다多 ，Спасибо ευχαριστὠ