Dispersive dam-break and lock-exchange flows in a two-layer fluid

TLDR: Dam break and lock exchange flows are considered in a Boussinesq two-layer fluid system in a uniform two-dimensional channel in this paper, where the focus is on inviscid 'weak' dam breaks or lock exchanges, for which waves generated from the initial conditions do not break, but instead disperse in a so-called undular bore.

Abstract: Dam-break and lock-exchange flows are considered in a Boussinesq two-layer fluid system in a uniform two-dimensional channel. The focus is on inviscid 'weak' dam breaks or lock exchanges, for which waves generated from the initial conditions do not break, but instead disperse in a so-called undular bore. The evolution of such flows can be described by the Miyata-Camassa-Choi (MCC) equations. Insight into solutions of the MCC equations is provided by the canonical form of their long wave limit, the two-layer shallow water equations, which can be related to their single-layer counterpart via a surjective map. The nature of this surjective map illustrates that whilst some Riemann-type initial-value problems (dam breaks) are analogous to those in the single-layer problem, others (lock exchanges) are not. Previous descriptions of MCC waves of permanent form (cnoidal and solitary waves) are generalised, including a description of the effects of a regularising surface tension. The wave solutions allow the application of a technique due to El's approach, based on Whitham's modulation theory, which is used to determine key features of the expanding undular bore as a function of the initial conditions. A typical dam-break flow consists of a leftwards-propagating simple rarefaction wave and a rightward-propagating simple undular bore. The leading and trailing edge speeds, leading edge solitary wave amplitude and trailing edge linear wavelength are determined for the undular bore. Lock-exchange flows, for which the initial interface shape crosses the mid-depth of the channel, by contrast, are found to be more complex, and depending on the value of the surface tension parameter may include 'solibores' or fronts connecting two distinct regimes of long-wave behaviour. All of the results presented are informed and verified by numerical solutions of the MCC equations.

Figures

Figure 7. Snapshots of numerical simulations of the four types of generalised lock exchanges with low surface tension (τ =0.05) (see text). Dotted lines show the theoretical predictions of the locations of the solibore, and the leading edge and trailing edge speeds of the undular bore/rarefaction waves. (a,c,e) Type I, initial conditions L− =L+ =−0.97, R− =0.99 (shear dominated), R+ =0.98 (buoyancy dominated). (g) Type II, L− = L+ =−0.97, R− =0.98 (SD), R+ =0.99 (BD). (b,d,f ) Type IV, L− =L+ =−0.97, R− =0.98 (BD), R+ =0.99 (SD). (h) Type III, L− =L+ =−0.97, R− =0.99 (BD), R+ =0.98 (SD).

Figure 3. Typical structure of the potential function −3P (h)/N (h) in the general, linear and solitary wave cases, corresponding to different values of the constants hi . Waves of permanent form exist only in potential wells (shaded). (a,c,e,g) Low surface tension regime τ =0.05. (b,d,f,h) High surface tension regime τ =0.09. (g) Potential function −3P (h)/N (h) for the case of a solibore (unique to the low surface tension case) and (h) the ‘Type II’ high surface tension situation.

Figure 4. (a) Development in the (x, t) plane of a simple rightward-propagating undular bore. Time reversibility in the MCC system implies that such a solution can be continued into t < 0, where the solution is a two-layer SWE simple wave of compression. (b) The typical development in the (x, t) plane of ‘dam-break’ initial conditions (v− = v+ =0, h+ <h− < 1/2). (c) The typical development in the (x, t)-plane of ‘lock-exchange’ initial conditions (v− = v+ =0, h+ < 1/2<h−). (d ) A (v, h) diagram illustrating how the resolved states are connected for the dam break initial conditions. (e) As (d ), but for the lock-exchange initial conditions.

Figure 8. (a, b) Snapshots of interface height at t =1152 illustrating the development of an undular bore (a) and rarefaction wave (b). Experimental parameters are as for figure 5, but with τ =0.09 (high surface tension regime). Dotted lines show the Whitham theory predictions. (c, d ) The Whitham theory predictions (solid lines) versus experimental results (squares) across a suite of experiments, as in figure 6, but for τ =0.09.

Figure 2. (a) Contour plot of the Riemann invariants R(v, h) (solid curves) and L(v, h) (dotted curves). (b) Possible right (+) and left (−) initial states for a simple leftward-travelling rarefaction waves in the buoyancy dominated (BD) and shear-dominated (SD) regimes. The end states must lie on contours of constant R(v, h) (solid curves). (c) The evolution of the interface height h(x, t) during the development of a rarefaction wave in the buoyancy dominated regime. The length of the arrows are proportional to the initial velocities in each layer. (d ) As (c), but for the shear-dominated regime.

Figure 5. (a,c,e) Snapshots of interface height from a numerical simulation of a rightwardpropagating undular bore with initial conditions L− =L+ =− √ 3/2, R+ = √ 3/2, R− =0.9701 and with low surface tension (τ =0.05). Dotted lines show the theoretical predictions of the locations of the leading solitary wave edge and trailing linear wave edge. (b, d, f ) Snapshots of a rightward-propagating rarefaction wave developed from ‘mirror image’ initial L− =L+ =− √ 3/2, R+ =0.9701, R− = √ 3/2, also for low surface tension (τ =0.05).

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Dispersivedambreakandlockexchangeflowsinatwo
layerfluid
J.G.ESLERandJ.D.PEARCE
JournalofFluidMechanics/Volume667/January2011,pp555585
DOI:10.1017/S0022112010004593,Publishedonline:14January2011
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J. Fluid Mech. (2011), vol. 667, pp. 555–585.
c

Cambridge University Press 2011
doi:10.1017/S0022112010004593
555
Dispersive dam-break and lock-exchange
flows in a two-layer fluid
J. G. ESLER† AND J. D. PEARCE
Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, UK
(Received 8 February 2010; revised 3 September 2010; accepted 5 September 2010)
Dam-break and lock-exchange flows are considered in a Boussinesq two-layer fluid
system in a uniform two-dimensional channel. The focus is on inviscid ‘weak’ dam
breaks or lock exchanges, for which waves generated from the initial conditions do
not break, but instead disperse in a so-called undular bore. The evolution of such
flows can be described by the Miyata–Camassa–Choi (MCC) equations. Insight into
solutions of the MCC equations is provided by the canonical form of their long wave
limit, the two-layer shallow water equations, which can be related to their single-layer
counterpart via a surjective map. The nature of this surjective map illustrates that
whilst some Riemann-type initial-value problems (dam breaks) are analogous to those
in the single-layer problem, others (lock exchanges) are not. Previous descriptions of
MCC waves of permanent form (cnoidal and solitary waves) are generalised, including
a description of the effects of a regularising surface tension. The wave solutions allow
the application of a technique due to El’s approach, based on Whitham’s modulation
theory, which is used to determine key features of the expanding undular bore as a
function of the initial conditions. A typical dam-break flow consists of a leftwards-
propagating simple rarefaction wave and a rightward-propagating simple undular
bore. The leading and trailing edge speeds, leading edge solitary wave amplitude and
trailing edge linear wavelength are determined for the undular bore. Lock-exchange
flows, for which the initial interface shape crosses the mid-depth of the channel, by
contrast, are found to be more complex, and depending on the value of the surface
tension parameter may include ‘solibores’ or fronts connecting two distinct regimes
of long-wave behaviour. All of the results presented are informed and verified by
numerical solutions of the MCC equations.
Key words: internal waves, ocean processes, shallow water flows
1. Introduction
A bore can be defined as a fluid flow connecting two uniform basic states.
In the context of single-layer flow, some of the most famous examples include
river bores such as those on the Severn (UK) and Dordogne (France). Large
amplitude bores tend to be turbulent, and turbulent bores are typically modelled by a
localised discontinuity satisfying physically or empirically derived Rankine–Hugionot
conditions. For relatively small depth ratios, however, the bore consists of a nonlinear
wavetrain and is said to be ‘undular’ (the early experimental work of Favre 1935
suggests a critical depth ratio for water waves of around 1.28).
† Email address for correspondence: gavin@math.ucl.ac.uk

556 J. G. Esler and J. D. Pearce
In this work, the focus is on internal undular bores generated by (partial) dam
breaks in a two-layer fluid. Dam breaks may be regarded as the canonical generation
mechanism for undular bores, and the dam-break problem is an essential pre-requisite
for understanding undular bores in more complex flows, such as those generated by
transcritical flow over topography (e.g. El, Grimshaw & Smyth 2009). The present
study is motivated by the ever increasing number of observations revealing the
ubiquity of internal undular bores in the atmosphere (e.g. Clarke 1998; Rottman &
Grimshaw 2002), where they may influence local weather events (e.g. thunderstorm
initiation), as well as in the coastal oceans (e.g. Holloway, Pelinovsky & Talipova
2001; Grimshaw 2002), where they have a role in mixing processes and in the energy
budgets of tidal flows.
The typical situation to be considered is shown in figure 1(a, b), which illustrates
schematically how a typical dam break initial condition evolves into a rightward-
propagating undular bore and leftward-propagating ‘rarefaction wave’. The somewhat
more complicated situation of a lock-exchange flow, which occurs in the two-layer
system but not in the single-layer one, is illustrated in figure 1(c, d). Understanding
how and why the development of the lock exchange differs from the dam break will
be a key focus below. An assumption made throughout the present work is that
the dam breaks considered are sufficiently weak, in the sense of the interface height
difference across the barrier being small (in some sense), and are initialised sufficiently
smoothly, that no wave breaking results. Based on this assumption, a suitable model
is the Miyata–Choi–Camassa (MCC) equations (following Miyata 1985; Choi &
Camassa 1999). The MCC equations are an extension of the two-layer shallow
water equations (SWE hereafter) with additional regularising dispersive terms which
preclude wave breaking entirely. Therefore, the results below are complementary
to previous analytical, experimental and numerical studies of dam breaks, lock
exchanges (e.g. Klemp, Rotunno & Skamarock 1997; Shin, Dalziel & Linden 2005) and
gravity currents (e.g. Benjamin 1968; Rottman & Simpson 1983; Klemp, Rotunno &
Skamarock 1994) in two-layer fluids. In the aforementioned studies, the focus has
been on ‘strong’ dam breaks, which include gravity currents as a special case, in which
wave breaking is observed (or assumed) and the internal bores are turbulent. There
has been much discussion in the literature about the appropriate Rankine–Hugoniot
conditions to apply at an internal bore or gravity current in a two-fluid system.
Possible jump conditions follow from the assumption of no energy dissipation in the
contracting layer (Chu & Baddour 1977; Wood & Simpson 1984), or the opposite
extreme that no energy dissipation occurs in the expanding layer. Theoretical (Li &
Cummins 1998) and experimental (Baines 1984; Rottman & Simpson 1989) results
indicate that spatially localised internal bores must propagate at speeds between the
values predicted by the above two hypotheses. In the present study, by contrast, we
will be concerned with undular bores which do not break and do not remain localised.
In single-layer flows the study of undular bores has followed Benjamin & Lighthill
(1954), who considered the steady undular bores that result when dissipation is present,
and Gurevich & Pitaevskii (1974), who solved a model ‘dam break’ initial-value
problem in the absence of dissipation. Both approaches exploit the assumption of
weak nonlinearity to derive solutions based on the cnoidal and solitary wave solutions
of the Korteweg–de Vries (KdV) equation. The dissipationless case can be argued to
be more fundamental, both as a description of the early time behaviour of an undular
bore before significant dissipation occurs, and as a generic paradigm relevant to many
branches of physics. The Gurevich–Pitaevskii solution is based on Whitham modula-
tion theory (following Whitham 1965), for which a scale separation is assumed between

Two-layer dam breaks 557
Upper layer:
density ρ
1
1–h
–
1–h
+
h
+
h
–
S
–
h
–
1–h
–
1–h
+
h
+
S
+
Dam break
(t = 0)
(a)
(b)
(c)
(d)
Dam break
(t > 0)
Lock
exchange
(t = 0)
Lock
exchange
(t > 0)
Lower layer:
density ρ
2
Upper layer:
density ρ
1
Expanding
rarefaction
wave
Expanding
undular
bore
Shear v
m
V
l
m
h
m
h
m
V
l
–
S
–
S
+
V
l
m
V
l
–
Lower layer:
density ρ
2
Upper layer:
density ρ
1
Lower layer:
density ρ
2
Exp.
R.W.
C
b
Solibore
Solibore
Shear v
m
Exp.
U.B.
Upper layer:
density ρ
1
Lower layer:
density ρ
2
C
b
Figure 1. (a)Dambreakt = 0: illustrating the physical set-up and step-like initial conditions
including definitions of the left and right interface heights h
−
and h
+
.(b)Dambreakt>0:
schematic illustration of a typical developing undular bore and rarefaction wave, including
the definition of the undular bore leading and trailing edge speeds s
+
and s
−
, the mid-state
(v, h)=(v
m
,h
m
) and long wave speeds V
l
+
and V
l
m
associated with the boundaries of the
rarefaction wave. (c) Lock exchange t = 0: the initial conditions for a lock exchange cross
the mid-depth of the channel. (d ) Lock exchange t>0: schematic illustration of a typical
lock-exchange development including solibores.

558 J. G. Esler and J. D. Pearce
the variation of mean quantities and a typical cnoidal wavelength. A consequence
of the integrability of the KdV equation is that the resulting modulation equations
may be obtained in Riemann invariant form and integrated to obtain full analytical
expressions for the variation of mean physical quantities across the undular bore.
Undular bores are generic features of nonlinear hyperbolic systems regularised by
(weak) dispersion, such as the MCC equations discussed above. Unlike the KdV,
however, the MCC equations, and their associated modulation equations, are not
thought to be integrable. Until recently, it was unclear how the Gurevich–Pitaevskii
approach could be usefully extended to non-integrable systems. However, El (2005)
(see also El, Khodorovskii & Tyurina 2003, 2005) have demonstrated that insights
from Whitham modulation theory can be used to determine details of undular bores
in a wide class of non-integrable hyperbolic equations supporting bidirectional wave
propagation. El’s (and co-authors’) technique allows simple undular bores to be
‘fit’ into solutions of the underlying long wave equations, much as localised jump
discontinuities are fit into solutions of the SWE as models of turbulent bores in the
classical dam-break problem.
A dissipationless undular bore differs from its localised turbulent counterpart in
that respect as it occupies a uniformly expanding region between a leading (e.g.
solitary wave) edge with speed s
+
and a trailing (e.g. linear wave) edge with speed s
−
,
as illustrated in figure 1. Using El’s technique, an explicit resolution of the details of
the undular bore is not necessary to obtain key quantities, such as the propagation
speeds of the boundaries of the undular bore (s
−
and s
+
), the leading edge solitary
wave amplitude and the trailing edge linear wavenumber. The application of El’s
technique to a wide class of systems supporting bidirectional wave propagation,
which will be shown below to include the MCC, is discussed in detail in El (2005,
§5), as well as in El et al. (2005). The main points of their arguments will be reviewed
below. A successful application of El’s technique to dam breaks in the single-layer
Su–Gardner (Su & Gardner 1969) dispersive shallow water model is described in
El, Grimshaw & Smyth (2006). Details of the single-layer dam-break solution were
essential pre-requisites for solution of the important physical problem of near-critical
flow over topography in the dispersive regime (El et al. 2009).
The objective of the present work is to apply El’s technique to two-layer dam breaks
and lock exchanges. In §2, the physical problem to be addressed is described together
with the MCC equations used to model the resulting fluid flows. The conservation
properties of the MCC are stated and the properties of the equations in the long
wave (SWE) limit are reviewed. A formal distinction is made between generalised ‘dam
break’ and ‘lock-exchange’ initial conditions. In §3, a general treatment of the steadily
propagating wave solutions of the MCC is presented. Previous results for solitary
wave speeds are generalised to allow for arbitrary vertical shears and the properties
of the ‘solibore’ solution of the MCC are reviewed. Section 4 considers the dam-break
problem, and it is demonstrated how El’s technique can be used to ‘fit’ undular bores
into solutions of the two-layer SWE. In §5, the predictions of El’s technique are
compared with numerical simulations of both dam-break and lock-exchange flows as
described by the MCC equations. Finally, in §6, conclusions are drawn.
2. A dispersive model for internal dam-break and lock-exchange flows
2.1. The physical problem
Figure 1 illustrates the initial conditions for the general physical situation of interest. A
uniform infinite two-dimensional channel of height H contains two fluids of densities

Frequently asked questions

Q1. What are the contributions in "Dispersive dam-break and lock-exchange flows in a two-layer fluid" ?

Link to this article: http: //journals. How to cite this article: J. G. ESLER and J. D. PEARCE ( 2011 ). 

Q2. What is the key step in finding the linear edge?

The key step in finding the linear (assumed here to be the trailing) edge involves the integration of the zero wave amplitude (a → 0), reduction of the Whitham system across the undular bore. 

Q3. What is the wavenumber at the trailing edge of the undular bore?

The wavenumber k− is the linear wavenumber at the trailing edge of the undular bore, and the speed of the trailing edge is given by the linear group velocity there s− = ∂(kc0)/∂k (k−, v−, h−). 

Q4. What are the generic features of undular bores?

Undular bores are generic features of nonlinear hyperbolic systems regularised by (weak) dispersion, such as the MCC equations discussed above. 

Q5. What is the simplest explanation for the asymmetric behaviour of the MCC equations?

It has been demonstrated that the long-time asymptotic behaviour of solutions of the MCC can be deduced using a technique due to El (2005, see also references therein), based on Whitham modulation theory, which allows undular bores to be ‘fit’ into solutions of the underlying long wave equations of a certain class of regularised nonlinear hyperbolic systems supporting bidirectional wave propagation. 

Q6. What is the amplitude of the solitary wave at the leading edge?

The amplitude of the solitary wave at the leading edge speed is determined by k̃+ and the speed of the upstream edge is given by the solitary wave speed s+ = cs(k̃+, v+, h+). 

Q7. What is the Whitham average of any function of (v, h)?

In the case of the (low surface tension) MCC the Whitham average of any function F (v, h) of baroclinic velocity, v, and interface height, h, can be written asF̄ = k2π ∫ 2π/k 0 F (v, h) dx = k√ 3π ∫ h3 h2 F (v(h), h) √ N(h)√ 

Q8. What is the way to write the conservation laws in ‘flux form’?

For the modulation theory presented in § 4, it is advantageous to write the conservation laws in ‘flux form’ using only the dependent variables (v, h) (see e.g. Kamchatnov 2000). 

Q9. What is the condition to guarantee N(h) > 0 throughout h?

A sufficient condition to guarantee N(h) > 0 throughout h ∈ [0, 1] can be obtained under the assumption that h1 and h4 do not coincide with the boundaries. 

Q10. What is the main difference between the two approaches?

Both approaches exploit the assumption of weak nonlinearity to derive solutions based on the cnoidal and solitary wave solutions of the Korteweg–de Vries (KdV) equation.