Dispersion in Shallow Water

Dispersion in Shallow Water

John D. CarterSeattle University

in collaboration with

Harvey SegurUniversity of Colorado at Boulder

David GeorgeU.S. Geological Survey

Diane HendersonPenn State University

John D. Carter Dispersion in Shallow Water

Outline

I. Experiments

II. St. Venant equations

III. KdV equation

IV. Serre equations

V. Whitham equation

VI. Summary


Experimental Set Up

Figure not to scale!

h0=10cm

Gauge a

x= 61cm

Gauge b

x= 561cm

Gauge c

x= 1061cm

Gauge d

x= 1561cm

Gauge e

x= 2061

Experiments conducted by Joe Hammack.


Experimental Initial Conditions

Figure not to scale!

h0=10cm

Gauge a

x= 61cm

Gauge b

x= 561cm

Gauge c

x= 1061cm

Gauge d

x= 1561cm

Gauge e

x= 2061

A0

Experiments conducted by Joe Hammack.


Parameter Definitions

I ε = Hh0

is a (dimensionless) measure of nonlinearity

I δ = h0λ is a (dimensionless) measure of shallowness

I H represents a typical wave height

I h0 represents the undisturbed water depth

I λ represents a typical wavelength


Experimental Measurements: A0 = 0.5cm344 J . L. H a m m k and H . Segur

0

- 0.03

- 0.06

- 0.09

0

- 0.03

- 0.06

-0.09 . mE 0

II i - 0.03

- 0.06

- 0.09

0

- 0.03

- 0-06

- 0.09

0

- 0.03

- 0.06 0 25 50 75 100 125 150 175 200 225

- X = t ( g / h ) f - x / h

FIGURE 2. Experimental wave systems: h = 10 cm, L, = 122 cm, A , = 0.5 cm, M = 0.0335, xo = - 3.66. (a) z/h = O or 37 = 0, (b ) z / h = 50 or 37 = 25, (c ) s /h = 100 or 37 = 50, (d) z /h = 150 or 37 = 75, (e) z / h = 200 or 37 = 100. -+, trajectory baaed on average wavenumber between two stations and linear dispersion relation; - +, extrapolation of previous trajectory.

3. Experiments 3.1. Equipment and procedure

The experimental equipment used in this study was described briefly in part 2 and in greater detail by Hammack (1973). The wave maker consists of a rectangular piston 61 cm long at the end of a wave tank. The rectangular wave propagating out of the generation region following a sudden downthrow of the piston has a length of twice the piston length and an amplitude of one-half the piston stroke. By varying the piston stroke while the quiescent water depth is fixed at h = 10cm, the nonlinearity of the initial wave is varied. For all of the experiments presented herein the length of the initial wave is constant at L, = 122 cm or L = 12.2. The initial wave amplitudes are A* = 0-5cm, 1.5cm and 2-5cm.

Waves are measured using parallel-wire resistance gauges and an oscillograph recorder a t the following positions: x = 0, 50h, 100h, 150h and 200h, where x = 0 is the downstream edge of the piston. Some of these measurements, especially those at x = 200h, are incomplete because the wave reflected from the downstream end of the tank returned prior to the complete passage of the rightward-running wave system. In the comparisons with theory, wave traces at xlh = 0, 50, 100, 150 and 200 are assumed to represent the spatial wave a t the times 37 = 0, 25, 50, 75 and 100, respectively, according to the normalization given by (2). Consequently, a rightward-

From J.L. Hammack and H. Segur, TheKorteweg-deVries equation and water waves.Part 3. Oscillatory waves, Journal of FluidMechanics, 84:337-358, 1978.

ε =0.5

10= 0.05

δ =10

2 ∗ 61= 0.08


Experimental Measurements: A0 = 1.5cmKorteweg-de Vries equation and water waves. Part 3 346

0.15

0

-0.15

-0.30

0.15 5

I1 z 0

-0.15

- 0.30

0.15

0

-0.15

-0.30

0.15

0

-0.15 0 25 50 75 100 125 150 175 200 225

- X = t (g/h)* - .y/h

FIG~RE 3. Experimental wave systems: h = 10 cm, L* = 122 cm, A , = 1.5 cm, M = 8.36 x 10-6, xo = -2.108. (a) z/h = 0 or 37 = 0, (a) z/h = 50 or 37 = 25, (c) z/h = 100 or 37 = 50, (d) z/h = 150 or 37 = 75, (e) z/h = 200 or 37 = 100. +, trajectory based on average wavenumber between two stations and linear dispersion relation ; - +, extrapolation of previous trajectory.

running wave necessarily appears leftward-running, i.e. the wave front appears a t the left in each figure. Moreover, a point moves to the left or right in succeeding measurements depending on whether its velocity is greater or less than (gh)B, respectively.

In the comparison of theory and experiment for the trailing wave region, wavenumbers k are required; however, only wave frequencies w = (h/g)Bw* are directly measurable from the experiments. To compute wavenumbers from the measured frequencies, the complete (all k) dispersion relation for linear water waves

w2 = k tanh k

is used. This procedure is adopted since measured frequencies and the water depth h = 10cm indicate that many of the oscillatory waves which evolve are not ‘long’ and considerable error is introduced by replacing (37) by its long-wave approximation.

(37)

3.2. Observed wave evolution

Figures 2-4 show the downstream wave measurements for three experiments with initial wave amplitudes of A , = 0.5 cm, 1.5 cm and 2.5 cm, respectively. The nor- malized wave amplitude f (or #y/h) is shown as a function of the non-dimensional co-ordinate - x (or t(g/h)t - x/h) . Wave traces are presented in this manner to empha- size tha t they are in fact temporal measurements at a fixed spatial location. which are

From J.L. Hammack and H. Segur, TheKorteweg-deVries equation and water waves.Part 3. Oscillatory waves, Journal of FluidMechanics, 84:337-358, 1978.

ε =1.5

10= 0.15

δ =10

2 ∗ 61= 0.08


Variable Definitions

I g represents the acceleration due to gravity

I h0 represents the undisturbed water depth

I h(x , t) represents the local water depth

I η(x , t) represents the free surface displacement

I Note: h(x , t) = h0 + η(x , t)

I u(x , t) represents the horizontal fluid velocity

I u(x , t) represents the depth-averaged horizontal fluid velocity


Initial Conditions for Numerics

h0=10cm

A0

x= 61cm


St. Venant Equations

The dimensional St. Venant (a.k.a. the classic shallow-waterequations) are

ht + (hu)x = 0

(hu)t +(1

2gh2 + hu2

)x

= 0

These equations are nondispersive.


St. Venant Equations Experiment #2: A0 = 0.5cm

St. Venant simulations computed by David George.


St. Venant Equations Experiment #3: A0 = 1.5cm

St. Venant simulations computed by David George.


KdV

The dimensional Korteweg-deVries equation is

ηt +√

gh0 ηx +3

2h0

√gh0 ηηx +

1

6h20√

gh0 ηxxx = 0

The linear dispersion relation is

ω =√

gh0(k − 1

6h20k

3)


Dispersion in KdV

0 1 2 3 4 5 6k h0

0.2

0.4

0.6

0.8

1.0

Ω

k c0

KdVEuler


KdV Experiment #2

10 20 30 40 50

-0.08

-0.06

-0.04

-0.02

0.00

HaL10 20 30 40 50

-0.08

-0.06

-0.04

-0.02

0.00

0.02HbL

Expt

KdV

20 40 60 80 100 120 140

-0.08-0.06-0.04-0.02

0.000.020.04

HcL

50 100 150 200

-0.06-0.04-0.02

0.000.020.04

HdL

50 100 150 200 250

-0.06-0.04-0.02

0.000.020.04

HeL


KdV Experiment #2, Larger t Interval

50 100 150 200

-0.08

-0.06

-0.04

-0.02

0.00

HaL100 200 300 400 500

-0.08

-0.06

-0.04

-0.02

0.00

0.02HbL

Expt

KdV

100 200 300 400 500

-0.08-0.06-0.04-0.02

0.000.020.04

HcL

200 400 600 800 1000

-0.06-0.04-0.02

0.000.020.04

HdL

200 400 600 800 1000

-0.06-0.04-0.02

0.000.020.04

HeL


KdV Experiment #3

10 20 30 40 50

-0.3

-0.2

-0.1

0.0

0.1HaL

20 40 60 80 100

-0.2

-0.1

0.0

0.1

HbLExpt

KdV

20 40 60 80 100 120 140

-0.15-0.10-0.05

0.000.050.100.15

HcL

50 100 150 200 250

-0.15-0.10-0.05

0.000.050.10

HdL

50 100 150 200 250

-0.10

-0.05

0.00

0.05

0.10HeL


KdV Experiment #3, Larger t Interval

100 200 300 400 500

-0.3

-0.2

-0.1

0.0

0.1HaL

100 200 300 400 500 600 700

-0.2

-0.1

0.0

0.1

HbLExpt

KdV

200 400 600 800 1000

-0.15-0.10-0.05

0.000.050.100.15

HcL

200 400 600 800 1000

-0.15-0.10-0.05

0.000.050.10

HdL

200 400 600 800 1000

-0.10

-0.05

0.00

0.05

0.10HeL


Serre

Serre story


Serre

The dimensional Serre equations are

ht + (hu)x = 0

ut + uux + ghx −1

3h

(h3(uxt + uuxx − (ux)2

))x

= 0


ω2 =3gk2h0

3 + k2h20


Dispersion in Serre

0 1 2 3 4 5 6k h0

0.2

0.4

0.6

0.8

1.0

Ω

k c0

SerreKdVEuler


Serre Experiment #2

10 20 30 40 50

-0.08

-0.06

-0.04

-0.02

0.00

0.02HaL

10 20 30 40 50

-0.08

-0.06

-0.04

-0.02

0.00

0.02HbL

Expt

Serre

20 40 60 80 100 120 140

-0.08-0.06-0.04-0.02

0.000.020.04

HcL

20 40 60 80 100 120 140

-0.06-0.04-0.02

0.000.020.04

HdL

20 40 60 80 100 120 140

-0.06-0.04-0.02

0.000.020.04

HeL


Serre Experiment #2

10 20 30 40 50

-0.08

-0.06

-0.04

-0.02

0.00

0.02HaL

20 40 60 80 100 120 140

-0.08

-0.06

-0.04

-0.02

0.00

0.02

HbLExpt

KdV

Serre

20 40 60 80 100 120 140

-0.08-0.06-0.04-0.02

0.000.020.04

HcL

20 40 60 80 100 120 140

-0.06-0.04-0.02

0.000.020.04

HdL

20 40 60 80 100 120 140

-0.06

-0.04

-0.02

0.00

0.02

0.04

HeL


Serre Experiment #2, Larger t Interval

100 200 300 400 500

-0.08

-0.06

-0.04

-0.02

0.00

0.02HaL

100 200 300 400 500

-0.08

-0.06

-0.04

-0.02

0.00

0.02HbL

Expt

Serre

200 400 600 800

-0.08-0.06-0.04-0.02

0.000.020.04

HcL

200 400 600 800

-0.06-0.04-0.02

0.000.020.04

HdL

200 400 600 800

-0.06-0.04-0.02

0.000.020.04

HeL


Serre Experiment #2

100 200 300 400 500

-0.08

-0.06

-0.04

-0.02

0.00

0.02HaL

100 200 300 400 500

-0.08

-0.06

-0.04

-0.02

0.00

0.02

HbLExpt

KdV

Serre

100 200 300 400 500

-0.08-0.06-0.04-0.02

0.000.020.04

HcL

100 200 300 400 500

-0.06-0.04-0.02

0.000.020.04

HdL

100 200 300 400 500

-0.06

-0.04

-0.02

0.00

0.02

0.04

HeL


Serre Experiment #3, Larger t Interval

100 200 300 400 500

-0.3

-0.2

-0.1

0.0

0.1HaL

200 400 600 800

-0.2

-0.1

0.0

0.1

0.2HbL

Expt

Serre

200 400 600 800

-0.15-0.10-0.05

0.000.050.100.15

HcL

200 400 600 800

-0.15-0.10-0.05

0.000.050.10

HdL

200 400 600 800

-0.10

-0.05

0.00

0.05

0.10HeL


Whitham

The dimensional Whitham equation is

ηt +3

2h0

√gh0 ηηx +

∫ ∞−∞

∫ ∞−∞

√gk tanh(kh0)

2πkeik(x−ξ)η

ξdk dξ = 0


ω =√

gk tanh(kh0)


Whitham Experiment #2

10 20 30 40 50

-0.08

-0.06

-0.04

-0.02

0.00

0.02HaL

20 40 60 80 100 120 140

-0.08

-0.06

-0.04

-0.02

0.00

0.02HbL

Expt

Whitham

20 40 60 80 100 120 140

-0.08-0.06-0.04-0.02

0.000.020.04

HcL

20 40 60 80 100 120 140

-0.06-0.04-0.02

0.000.020.04

HdL

20 40 60 80 100 120 140

-0.06-0.04-0.02

0.000.020.04

HeL


Whitham Experiment #2, Larger t Interval

100 200 300 400 500

-0.08

-0.06

-0.04

-0.02

0.00

0.02HaL

100 200 300 400 500

-0.08

-0.06

-0.04

-0.02

0.00

0.02HbL

Expt

Whitham

100 200 300 400 500 600

-0.08-0.06-0.04-0.02

0.000.020.04

HcL

100 200 300 400 500 600

-0.06-0.04-0.02

0.000.020.04

HdL

100 200 300 400 500 600

-0.06-0.04-0.02

0.000.020.04

HeL


Whitham Experiment #3, Larger t Interval

100 200 300 400 500 600

-0.3

-0.2

-0.1

0.0

0.1HaL

100 200 300 400 500 600

-0.2

-0.1

0.0

0.1

0.2

HbLExpt

Whitham

100 200 300 400 500

-0.15-0.10-0.05

0.000.050.100.15

HcL

100 200 300 400 500 600

-0.15-0.10-0.05

0.000.050.100.15

HdL

100 200 300 400 500 600

-0.10

-0.05

0.00

0.05

0.10

HeL


Summary

1. Dispersion can be important in shallow water

2. “Any old form” of dispersion is not necessarily sufficient

3. Nonlinear effects are important in shallow water

4. None of the examined models get the amplitudes correct

5. Dissipation likely plays an important role


Dispersion in Shallow Water

Documents

Transcript of Dispersion in Shallow Water