dispersive dam breaks and lock-exchanges in a two-layer fluid · introduction the boussinesq...

IntroductionThe Boussinesq two-layer fluid

Dam breaks and lock exchanges

Dispersive dam breaks and lock-exchanges ina two-layer fluid

Gavin Esler and Joe Pearce

Department of MathematicsUniversity College London

BIRS: Dispersive hydrodynamics

May 21, 2015

Gavin Esler and Joe Pearce Internal dam breaks and lock-exchanges



The Physical ProblemUndular bores in the atmosphere and ocean

Two-layer problem: initial conditions

Consider a ‘lock release’ in a Boussinesq two-layer fluid.Waves do not break: No turbulent internal bores.Waves can be strongly nonlinear, but are long (kH . 1).





Undular bore over the Gulf of Mexico

(Loading ub.avi)


ub.aviMedia File (video/avi)




Single layer problem

Serre-Su-Gardner-Green-Naghdi equations.El, Grimshaw, Smyth (Phys. Fluids, 2006).Generic behaviour: Rightwards-propagating undularbore, leftwards rarefaction wave.





Two-layer problem

Will the fluid behave as the single-layer system?How are the details determined...?




Equation set: The Miyata-Choi-Camassa equationsThe long wave limit: Two-layer shallow waterWaves in the MCC equations

Miyata-Choi-Camassa Equations - I

The following nondimensional extended shallow water set1 canbe used to model internal undular bores:

u1t + u1u1x = −Πx + 13h1(

h31G[u1])

x+ τ2 h1xxx

u2t + u2u2x = −Πx − h2x + 13h2(

h32G[u2])

x+ τ2 h2xxx

h1t + (u1h1)x = 0,h2t + (u2h2)x = 0,

ui velocity, hi thickness in i th layer. Π pressure at rigid lid.G[·] is a nonlinear differential operator, acting on any f (x , t),

G[f ] = fxt + ffxx − (fx )2.τ is a Bond number - a measure of the surface tension.

1Miyata 1985; Choi & Camassa 1999.Gavin Esler and Joe Pearce Internal dam breaks and lock-exchanges




Miyata-Choi-Camassa Equations - II

In the absence of topography, the MCC equations are just twocoupled PDEs in disguise.

1 Work in the zero-total momentum frame withu1h1 + u2h2 = 0.

2 Use h1 + h2 = 1 to define h = h2 with h1 = 1− h.3 Introduce a baroclinic velocity v = u2 − u1.

Then the MCC equations can be written

ht + (vh(1− h))x = 0,vt +

(12v

2(1− 2h) + h)

x= τhxxx + 13h

(h3G[v(1− h)]

)x

− 13(1−h)(

(1− h)3G[−vh])

x.

Only two dependent variables (v ,h) remain.Gavin Esler and Joe Pearce Internal dam breaks and lock-exchanges




Dispersionless limit

For long waves (k � 1) neglect dispersion and surface tension

ht + (vh(1− h))x = 0,vt +

(12v

2(1− 2h) + h)

x= 0.

The corresponding single layer shallow water equations are

σt + (uσ)x = 0,

ut +(

12u

2 + σ)

x= 0,

for depth σ and velocity u.





Exact Correspondence...!?

Both systems can be mapped into the following form2:[∂

∂t+(3

4L +14R) ∂∂x

]L = 0,

[∂

∂t+(3

4R +14L) ∂∂x

]R = 0,

where L and R are the left and right Riemann invariants:

2-layer

L = v(1− 2h)− 2√

h(1− h)(1− v2)R = v(1− 2h) + 2

√h(1− h)(1− v2)

∣∣∣∣∣∣∣∣

1-layer

L = u − 2√σR = u + 2

√σ

The dynamics of the 2-layer and 1-layer SWE appearidentical...!?

2E.g. Cavanie, 1969; Chumakova et al. 2009Gavin Esler and Joe Pearce Internal dam breaks and lock-exchanges




Two-layer SWE: Riemann invariants

Contours: L(v ,h) (dotted)Contours: R(v ,h) (solid).

The hyperbolic region of (v ,h) space is [−1,1]× [0,1].In general four points in the region have the same value of(L,R).





Two-layer SWE: ‘St Andrew’s Cross’ mapping

1

1−layer

u

v

2−layer

h

0

1

−1 1 1−1

SD

BD

BD

SD

σ

Mapping 2-layer SWE→ 1-layer SWE is surjective.Four quadrants in the 2-layer (v ,h) domain→ a triangularsubset of the 1-layer (u, σ) domain.Physical reasoning =⇒ the existence of an up-downsymmetry h→ 1− h, v → −v .Less-obvious symmetry: h→ 1− v/2, v → 1− 2h.





MCC equations: Steadily propagating waves

Wave-like solutions of the MCC (steady propagation at speedc) can be sought: use3

∂t → −c∂xThe mass equation can be integrated to eliminate v and themomentum equation integrated twice to get the potential form:

(hx )2 =3P(h)N(h)

,

P(h) = (h − h1)(h − h2)(h − h3)(h − h4)N(h) = h1h2h3h4(1 − h) + (1 − h1)(1 − h2)(1 − h3)(1 − h4)h − 3τh(1 − h)

Steady wave solutions are entirely described by the roots hi .

3following Choi and Camassa, 1999.Gavin Esler and Joe Pearce Internal dam breaks and lock-exchanges




MCC equations: Steadily propagating waves

Interface height h

Waves can be understood us-ing the potential function.





MCC equations: The Solibore

18 W. Choi and R. Camassa

h1

h2

h1

ʹ′

h2

ʹ′

u1

ʹ′

u2

ʹ′

q1

q2

u1

= c

u2

= c

Figure 9. Steady flow in a frame moving with a left-going front.

where ζ0 ≡ ζ(X = 0) = am/2,

κm =

��3g(ρ2 − ρ1)

c2m(ρ1h21 − ρ2h

22)

��1/2

, (3.71)

and the function Xf is

Xf(x, y, z) = (y − z)1/2 log

(x − z)1/2 + (y − z)1/2

(x − z)1/2 − (y − z)1/2− (−z)1/2 log

(x − z)1/2 + (−z)1/2

(x − z)1/2 − (−z)1/2.

(3.72)

As figure 8 shows, the limiting wave form is a front slowly varying from 0 to am.We remark that for the case of a homogeneous layer, the analogue of the present

fully nonlinear model is offered by the GN system given by (3.20) and (3.22) withP = 0. Its solitary wave solutions are

ζ(X) = a sech2(κ1X), κ21 =

3a

4(1 + a), c

2 = 1 + a , (3.73)

and it is interesting to notice that no limiting wave amplitude exists in this case.Because we have not imposed any assumption on the magnitude of the wave

amplitude in order to derive our long-wave model, and since the solution (3.70)–(3.72) is consistent with the long-wave assumption (3.1), it is natural to expect that asimilar front solution exists for the full Euler system. In fact, this was demonstratedby Funakoshi & Oikawa (1986), who also used their proof to validate their numericalsolutions. For completeness, we now come back to the existence proof of Euler frontsolutions (rederiving the result of Funakoshi & Oikawa 1986) and compare these tothe highest wave solutions of the present long-wave model.

3.4. Internal bore: exact theory

We assume the set-up illustrated in figure 9 for an internal bore moving from rightto left at constant speed c into a two-layer stratified fluid at rest at x = −∞. Thus,in a frame moving with such a left-going front of speed c, the velocity at x = −∞is c in both fluids, i.e. u1 = u2 = c. On the other hand, the velocity at x = ∞ can beexpected to be different, say u�1 (u

�

2), in an upper (lower) fluid layer whose thicknessis h�1 (h

�

2). The question of existence of front-like solutions is then equivalent to thatof finding c, u�

iand h�

ifor given ρi and hi, i = 1, 2, such that all three basic physical

conservation laws of mass, momentum and energy hold.Mass conservation in each fluid implies

ch1 = u�

1h�

1, ch2 = u�

2h�

2, (3.74)

Solibores connect states with equal (L,R)

Solibore speed : cb = 12 +14(L + R).




Whitham modulation theoryDam breaksLock exchanges

Riemann problem

Consider general step-like initial conditions for the MCCequations

v(x ,0) ={

v− x < 0v+ x > 0

h(x ,0) ={

h− x < 0h+ x > 0

Consider in particular states subject to a further relationship

Φ(v−,h−, v+,h+) = 0,

so that only a rightwards propagating undular bore (norarefaction wave) emerges at long times.





Whitham-El method

The MCC equations are not integrable.

Whitham modulation theory can be used to obtainWhitham equations from conservation laws.

Whitham-El method∗ can be used to ‘fit’ dispersive shocksinto solutions of non-dispersive system.

∗See El (2005), El et al. (2006, Phys. Fluids) and Esler and Pearce (JFM,2011) for details.





Whitham Modulation Equations

Use Whitham averaging:

F̄ =k

2π

∫ 2π/k

0F (v ,h) dx =

k√3π

∫ h3h2

F (v(h),h)√

N(h)√P(h)

dh,

F̄ = F̄ (h1,h2,h3,h4) modulation variables. Apply toconservation laws

(P̄i)t + (Q̄j)x = 0, (P1,P2,P3,P4) = (M, I, E , k)(Q1,Q2,Q3,Q4) = (V,J ,F , ck)

Result: Four first-order PDEsyt + Byx = 0,

where yT = (h1,h2,h3,h4) and B = P−1Q, with P and Q 4× 4matrices with components pij = ∂P̄i/∂hj , qij = ∂Q̄i/∂hj .





Time-reversibility argument

The MCC system is timereversible.

Characteristics of theWhitham system arecontinuous across t = 0

=⇒ that a undular boreevolves into a rarefactionwave in -ve time.

=⇒ rightward-propagating undular bore satisfies samejump condition as a -ve time rarefaction wave: L− = L+.





Expansion fan solution

For our step-like initial conditions a similarity solution y(x/t)can be sought. Obtain

(B−

( xt

)I)

y′ = 0.

Deduce that the similarity solution must be (schematically) ofthe form

Fi(h1,h2,h3,h4) = 0, λj(h1,h2,h3,h4) =xt, i = 1,2,3.

for unknown functions Fi , and where λj is one of theeigenvalues of B.

But what next...?





Whitham-El method: Matching conditions

Linear (trailing) edge: At the linear edge (a→ 0) of theundular bore the Whitham system reduces to:

a = 0 : h̄t +(v̄ h̄(1− h̄)

)x = 0,

v̄t +(

12 v̄

2(1− 2h̄) + h̄)

x= 0.

kt +(kc0(v̄ , h̄, k)

)x = 0

One of the eigenvalues of the reduced system must equals− = x/t at the linear edge. Deduce that

s− =∂(kc0)∂k

(k−, v−,h−)

It remains to determine the trailing edge wavenumber k−.





Whitham-El method: Linear edge

How can the trailing wavenumber k− be determined?

Key Insight: The a = 0 reduction of the Whitham system canbe integrated across the undular bore. (Intersects actualsolution at x/t = s±). Integrate

Rt +(3

4R +14L±

)Rx = 0, kt + (c0(k ,L±,R)k)x = 0.

After some considerable working, get the first order ODE:d(k2)

dR=

34 R +

54 L − 2c

r0 + 2k

2[HR(

13 (c

r 20 + 2H) − τ − 148 (R − L)

2)− 124 H(R − L)

]k−2(V r − cr0)(R + L − 2(1 +

13 k

2H)cr0) − 2H(13 (c

r 20 + H) − τ −

148 (R − L)2)

,

with the boundary condition k(R+) = 0.





Whitham-El method: Solitary wave edge

Solitary wave amplitude a+ and solitary wave speed s+obtained analogously. Use conjugate wavenumber k̃ .

k̃ =√

3π

(∫ h2h1

√N(h)/

√−P(h) dh

)−1

to parametrise a.





Dam break: Resolution of initial data

Stationary initial conditions lead to a rightwards undularbore and a leftwards rarefaction wave.The mid-state is defined by (Lm,Rm) = (L+,R−).





Dam break: Resolution of initial data - II





Dam break: Resolution of initial data - III





Dam break: numerical results- I

Distance x

Initial conditions: (L−,R−) = (−0.8660,0.9701),Initial conditions: (L+,R+) = (−0.8660,0.8660).Linked with a tanh-function (Half-width 2H).Periodic domain (400H). 2048 Fourier modes.





Dam break: numerical results- II

Distance x

Resolves into rightward propagating undular bore only.Edges of the undular bore: predicted from theory (dottedlines).





Dam break: numerical results- III

Distance x

Agreement improves at later times.





Dam break: Summary of edge speed predictions

Interface height h−

Theoretical predictions are good for small step-sizes, butbecome poorer as R− → 1 (h− →≈ 0.35).





Lock exchange: Resolution of initial data

Solutions can be constructed from ‘solibores’ and undularbores / rarefaction waves.Correct causality: solibores connect to left and right (notmid-)states.





Lock exchange: (x , t)-plane

Solibores propagate ahead of undular bore / rarefactioninto undisturbed fluid.





Lock exchange: Resolution of initial data - II





Lock exchange: Resolution of initial data - III





Lock exchange: numerical results- I

Distance x

Initial conditions: (L−,R−) = (−0.97,0.99),Initial conditions: (L+,R+) = (−0.97,0.98).Rightward propagating solibore followed by undular bore.





Lock exchange: numerical results- II

Distance x

Predicted positions: dotted lines.Asymptotic solution takes relatively long to emerge.





Lock exchange: numerical results- III

Distance x

A constant ‘image’ state emerges between solibore andundular bore.





The Gardner equation

16

−200 −100 0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1Analytical Region 2

x

u

(a)

−200 −100 0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1Numerical Region 2

x

u

(b)

FIG. 11: Decay of an initial discontinuity for the Gardnerequation with α = 1. The step parameters are u− = 0.7,u+ = 0.1 (Region 2, {u− ← UB (u∗) SB → u+}). (a)analytical solution in the form of a modulated periodic wave;(b) numerical solution; the analytically found edges of theundular bore and solibore are shown by dashed lines. Bothplots correspond to t = 300.

One can see from (114) that for u− = 1/2α the speed ofthe solibore coincides with the speed of the leading soli-ton in the undular bore so the “borderline” wave patternin Fig. 12b can be interpreted in both ways: as a normal,“bright”, undular bore or, equivalently, as a reversed,“dark”, undular bore with an attached solibore. Whenwe increase u−, the solibore separates from the undularbore, which in its turn acquires the reversed waveformwith a distinct dark soliton structure near the leadingedge (see Fig. 12d).

Region 3, 1/α−u− < u+ < 12α < u−, u−+u+ > 1/α.{u− ← RW (u∗) SB → u+}

This region is analogous to Region 2, since the valuesu− and u+ again lie in different domains of monotonic-ity of the function w(u), thus a single-wave resolutionis not possible. However, now the intermediate stateu∗ = 1/α − u+ < u+ (see Fig. 10b) so a reversed rar-efaction wave is generated instead of reversed undularbore. The solution for the rarefaction wave is given byformula (80), where ul = u

− and ur = u∗. The soliboresolution connecting u∗ and u+ is the same as in Region2. The numerical plot is presented in Fig. 13a alongwith the analytically found positions of the solibore andthe rarefaction wave boundaries. We note that, sincemaxu+(1 − αu+) = 14α , the speed of the solibore sk =

50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

x

u

(a)

50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

x

u

(b)

50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

x

u

(c)

50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

x

u

(d)

FIG. 12: The transformation of the undular bore structurefrom the normal, “bright soliton”, pattern in Region 1 (plot1) to the reversed, “dark soliton”, pattern in Region 2 (plot 4).Numerical simulations of the Gardner equation with α = 1.The downstream state, u+ = 0.1, is the same for all cases, theupstream state u− is taken in the range u− = 0.49 < 1/2α(Region 1) to u− = 0.58 > 1/2α (Region 2)

1α +2u

+(1−αu+) is always greater than that of the rightedge of the rarefaction wave, s+ = sr = 6u

+(1 − αu+)(see (81)). At the boundary between Regions 3 and 4,when u+ = 1/(2α), we have sk = sr and the solibore gets“attached” to the right edge of the rarefaction wave.

Region 4, 1/(2α) ≤ u+ < u−. {u− ← RW u+}.A single reversed rarefaction wave is produced. It is

Model equation:

ut +6uux−6αu2ux +uxxx = 0

Whitham equations areintegrable.

Solutions fully classified.

Kamchatnov et al.(Phys. Rev. E, 2012)





Conclusions

The MCC equations can be used to study dam-breaks andlock-exchanges in a 2-layer fluid.Whitham-El method can be used to ‘fit’ dispersive undularbores into 2-layer SWE solutions.A clear distinction can be made between ‘dam-break’ and‘lock-exchange’ flows.All phenomenology seen in the Gardner equation isobserved.


dispersive dam breaks and lock-exchanges in a two-layer fluid · introduction the boussinesq...

Documents