Dispersion in Shallow Water

Transcription

1 Seattle University in collaboration with Harvey Segur University of Colorado at Boulder David George U.S. Geological Survey Diane Henderson Penn State University

2 Outline I. Experiments II. St. Venant equations III. KdV equation IV. Serre equations V. Whitham equation VI. Summary

3 Experimental Set Up Figure not to scale! x 61cm x 561cm x 1061cm x 1561cm 1 h 0 10cm Experiments conducted by Joe Hammack.

4 Experimental Initial Conditions Figure not to scale! x 61cm x 561cm x 1061cm x 1561cm 1 h 0 10cm 0 Experiments conducted by Joe Hammack.

5 Parameter Definitions ɛ = H h 0 δ = h 0 λ is a (dimensionless) measure of nonlinearity is a (dimensionless) measure of shallowness H represents a typical wave height h 0 represents the undisturbed water depth λ represents a typical wavelength

6 Experimental J. L. Hammk Measurements: and H. Segur A 0 = 0.5cm me 0 II i X=t (g/h)f-x/h xperimental From wave J.L. systems: Hammack h = and 10 cm, H. Segur, L, = 122 Thecm, A, = 0.5 cm, M = , (a) z/h = O or Korteweg-deVries 37 = 0, (b) z/h equation = 50 or 37 and= water 25, (c) waves. s/h = 100 or 37 = 50, (d) z/h = 5, (e) z/h = Part 200 or Oscillatory = , waves, trajectory Journal baaed of Fluid on average wavenumber between and linear dispersion Mechanics, relation; 84: , - +, extrapolation of previous trajectory. ɛ = = 0.05 δ = = 0.08 ents

7 Experimental Measurements: A 0 = 1.5cm Korteweg-de Vries equation and water waves. Part z 0 I X = t (g/h)*-.y/h. Experimental wave systems: h = 10 cm, L* = 122 cm, A, = 1.5 cm, M = 8.36 x (a) From z/h = J.L. 0 or Hammack 37 = 0, and (a) H. z/h Segur, = 50 The or 37 = 25, (c) z/h = 100 or 37 = 50, 150 or 37 = 75, Korteweg-deVries (e) z/h = 200 or equation 37 = 100. and+, water trajectory waves. based on average wavenumber wo stations Part and linear 3. Oscillatory dispersion waves, relation Journal ;- +, of Fluid extrapolation of previous trajectory. Mechanics, 84: , ɛ = = 0.15 δ = = 0.08 wave necessarily appears leftward-running, i.e. the wave front appears at in each figure. Moreover, a point moves to John the D. left Carter or right Dispersion in succeeding Shallow Water

8 Variable Definitions g represents the acceleration due to gravity h 0 represents the undisturbed water depth h(x, t) represents the local water depth η(x, t) represents the free surface displacement Note: h(x, t) = h 0 + η(x, t) u(x, t) represents the horizontal fluid velocity ū(x, t) represents the depth-averaged horizontal fluid velocity

9 Initial Conditions for Numerics x 61cm A 0 h 0 10cm

10 St. Venant Equations The dimensional St. Venant (a.k.a. the classic shallow-water equations) are h t + (hu) x = 0 (hu) t + ( 1 2 gh2 + hu 2) x = 0 These equations are nondispersive.

11 St. Venant Equations Experiment #2: A 0 = 0.5cm St. Venant simulations computed by David George.

12 St. Venant Equations Experiment #3: A 0 = 1.5cm St. Venant simulations computed by David George.

13 KdV The dimensional Korteweg-deVries equation is η t + gh 0 η x + 3 gh0 ηη x + 1 2h 0 6 h2 0 gh0 η xxx = 0 The linear dispersion relation is ω = gh 0 ( k 1 6 h2 0k 3)

14 Dispersion in KdV Ω Euler KdV

15 KdV Experiment #

16 KdV Experiment #2, Larger t Interval

17 KdV Experiment #

18 KdV Experiment #3, Larger t Interval

19 Serre Serre story

20 Serre The dimensional Serre equations are h t + (hū) x = 0 ū t + ūū x + gh x 1 ( h 3( ū xt + ūū xx (ū x ) 2)) 3h = 0 x The linear dispersion relation is ω 2 = 3gk2 h k 2 h 2 0

21 Dispersion in Serre Ω Euler KdV Serre

22 Serre Experiment #

23 Serre Experiment # Expt KdV 0.08 c d e

24 Serre Experiment #2, Larger t Interval

25 Serre Experiment # Expt KdV 0.08 c d e

26 Serre Experiment #3, Larger t Interval

27 Whitham The dimensional Whitham equation is η t + 3 gk tanh(kh0 ) gh0 ηη x + e ik(x ξ) η 2h 0 2πk ξ dk dξ = 0 The linear dispersion relation is ω = gk tanh(kh 0 )

28 Whitham Experiment #

29 Whitham Experiment #2, Larger t Interval

30 Whitham Experiment #3, Larger t Interval

31 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