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Journal of Hydraulic Research Vol. 54, No. 5 (2016), pp. 541–557

http://dx.doi.org/10.1080/00221686.2016.1174961

© 2016 International Association for Hydro-Environment Engineering and Research

Entrainment and mixing in unsteady gravity currents

LUISA OTTOLENGHI, Postdoctoral Researcher, Department of Engineering, University Roma Tre,

via Vito Volterra 62, 00146, Rome, Italy

CLAUDIA ADDUCE (IAHR MEMBER), Associate Professor, Department of Engineering, University Roma Tre, via Vito

Volterra 62, 00146, Rome, Italy

Email: claudia.adduce@uniroma3.it (author for correspondence)

ROBERTO INGHILESI, PhD Student, Department of Engineering, University Roma Tre, via Vito Volterra 62, 00146, Rome, Italy;

Institute for Environmental Protection and Research, via Vitaliano Brancati 48, 00144, Rome, Italy

VINCENZO ARMENIO (IAHR MEMBER), Full Professor, Department of Engineering and Architecture, University of Trieste,

Piazzale Europa 1, 34127, Trieste, Italy

FEDERICO ROMAN, R & D Engineer, Ieﬂuids S.r.l., Piazzale Europa 1, 34127, Trieste, Italy

ABSTRACT

Entrainment and mixing in lock-exchange gravity currents are investigated by large eddy simulations. Nine cases are analysed, varying the initial

excess density driving the motion and the aspect ratio rof the initial water depth to the lock length. Laboratory experiments are also performed and

a fair agreement between numerical simulations and measurements is found. Mixing between the gravity current and the ambient ﬂuid, in both the

slumping and self-similar phases, is studied for a range of entrainment parameters, gravity current fractional area and using an energy budget method.

The entrainment is found to increase as rdecreases. The occurrence of irreversible mixing is detected during the entire development of the ﬂow, i.e.

both in the slumping and self-similar phases. A higher amount of mixing is observed as rdecreases and the initial excess density increases.

Keywords: Buoyancy-driven ﬂows; density currents; entrainment; laboratory studies; large eddysimulations; lock-release; mixing

1 Introduction

Gravity currents are buoyancy-driven ﬂows, in which the den-

sity gradient drives the motion mainly in the horizontal direc-

tion. The diﬀerence in ﬂuid density derives from temperature

or concentration gradients. When the diﬀerence in density is

caused by the presence of particulate matter in the body of

the current, the ﬂow is commonly known as turbidity current.

Gravity currents can be generated either by natural or anthro-

pogenic causes. Examples of gravity currents are sea breeze

fronts, oceanic overﬂows, avalanches, sand storms, oil spillages

and pollutant discharge in water bodies (Simpson 1997).

Gravity currents have been widely investigated by both lab-

oratory experiments and numerical simulations. Steady and

unsteady gravity currents are usually studied in the laboratory by

releasing a dense ﬂuid into a lighter one, by a continuous source

or by the lock-exchange technique (Cenedese & Adduce 2008;

Laanearu, Cuthbertson, & Davies 2014; La Rocca, Prestininzi,

Adduce, Sciortino, & Hinkelmann 2013). The lock-exchange

gravity current is produced by a sudden release of a ﬁxed vol-

ume of ﬂuid into another one of a diﬀerent density. This current,

among the others, is representative of a fast discharge of a dense

gas in the atmosphere, of the interaction of fresh and salt water

during the opening of a gate in a navigation channel near the

coast, and of the mobilization of sediments caused by subma-

rine landslides (Kneller & Buckee 2000). The lock-exchange

laboratory experiment consists in ﬁlling a tank divided by a slid-

ing gate with two ﬂuids at diﬀerent densities. When the gate is

removed the two ﬂuids interact with each other, generating a

horizontal pressure gradient which drives the motion. The heav-

ier ﬂuid starts to move along the bottom of the tank beneath

the ambient ﬂuid, mixing with it and a head, a body and a tail

region can be distinguished (Cantero, Lee, Balachandar, & Gar-

cía 2007; Cantero, Balachandar, García, & Bock 2008). Two

Received 30 July 2015; accepted 3 April 2016/Open for discussion until 30 April 2017.

ISSN 0022-1686 print/ISSN 1814-2079 online

http://www.tandfonline.com

541

542 L. Ottolenghi et al. Journal of Hydraulic Research Vol. 54, No. 5 (2016)

kinds of instabilities characterize the ﬂow: Kelvin–Helmholtz

billows and lobe-and-cleft structures. The ﬁrst one is caused by

the velocity shear at the interface between the two ﬂuids; the

second one is related to the intrusion of ambient ﬂuid under the

nose of the dense current.

According to the shallow water theory, diﬀerent phases in

the development of a gravity current can occur (Rottman &

Simpson 1983): ﬁrst, a slumping phase characterized by con-

stant velocity of the front develops; it is followed by a self-

similar phase, where inertial and buoyancy forces dominate

the dynamics and during which the front of the current slows

down; a third phase is observed when the viscous forces become

signiﬁcant (Huppert 1982).

Studies of gravity currents by means of high-resolution

numerical models which solve the three-dimensional Boussinesq

form of the Navier–Stokes equations are relatively recent.

Speciﬁcally, direct numerical simulations (DNS) and large eddy

simulations (LES) have been employed to study lock-exchange

gravity currents in diﬀerent settings (Cantero et al. 2007,2008;

Dai 2013,2015; Härtel, Meiburg, & Necker 2000;Ooi,

Constantinescu, and Weber 2009; Tokyay, Constantinescu, and

Meiburg 2011,2012). These studies have established that DNS-

LES are reliable and accurate methodologies for the analysis of

the physics of gravity currents.

Since the dynamics of gravity currents are aﬀected by the

amount of the entrained ambient ﬂuid, diﬀerent studies have

been focused on this issue. The entrainment was parametrized

as a function of the Froude number by Turner (1986) and

Parker, Garcia, Fukushima, and Yu (1987), based on laboratory

experiments of steady gravity currents. Steady gravity currents

ﬂowing down a sloping bottom in a rotating ﬂuid were anal-

ysed in the studies of Cenedese and Adduce (2008,2010) and

the dependence of the entrainment parameter on both Froude

and Reynolds numbers was discussed. On the other side, the

evaluation of mixing and entrainment in unsteady gravity cur-

rents is still an open issue. Previous investigations, devoted to

mixing in unsteady gravity currents, led to diﬀerent conclusions

regarding the presence of mixing during the slumping phase:

Hallworth, Huppert, Phillips, and Sparks (1996) observed mix-

ing only after the transition to the self-similar phase, while in the

experiments of Hacker, Linden, and Dalziel (1996), the occur-

rence of mixing was detected also during the slumping phase.

In Hacker et al. (1996), mixing was estimated by the analysis

of the iso-density levels, measured by a light attenuation tech-

nique. In the experiments of Hallworth et al. (1996), an alkaline

gravity current was released into an acidic ambient and the visu-

alization of the mixing phenomena was allowed by the use of

a coloured pH indicator. According to Fragoso, Patterson, and

Wettlaufer (2013), contrasting results on the occurrence of mix-

ing in the slumping phase must be attributed to the experimental

method of Hallworth et al. (1996), which could only discrimi-

nate between the homogeneous head and the diﬀusive wake of

the gravity current. Since this method was not able to detect

small values of density diﬀerences in the head region, diﬀerent

threshold values were probably considered to deﬁne the inter-

face between the dense and the ambient ﬂuids, aﬀecting the

entrainment evaluation. The entrainment evaluation is found to

be aﬀected by the choice of the density threshold used to deﬁne

the interface between the dense current and the ambient ﬂuid

(Hacker et al. 1996). Therefore an energetic approach, which

makes use of the variation of the background potential energy

to mark the presence of irreversible mixing (Winters, Lombard,

Riley, & D’Asaro 1995) and which is not dependent on density

thresholds, is a suitable method to investigate the occurrence

of mixing in gravity currents (Fragoso et al. 2013). Simpliﬁed

models, such as two-layer shallow water ones, being not able to

solve the small scale mixing processes, need a suitable entrain-

ment parametrization to correctly simulate the dynamics of a

lock-release gravity current (Adduce, Sciortino, & Proietti 2012;

Lombardi, Adduce, Sciortino, & La Rocca 2015; Ross, Dalziel,

and Linden 2006). In this context the entrainment parametriza-

tion for unsteady gravity currents is still uncertain. In addition,

to the authors’ knowledge there is a lack of study on entrainment

and mixing in gravity currents, which makes use of diﬀerent

methods with results consistent among them. Then, although

lock-exchange gravity currents have been widely investigated,

there are still open issues, deserving more research.

In this paper, unsteady gravity currents generated by the

lock-exchange technique are analysed through numerical wall-

resolving LES which solve the Boussinesq form of the three-

dimensional Navier–Stokes equations. The novel contribution

of this study is the evaluation of entrainment and mixing in

unsteady gravity currents, focusing on the eﬀect of the aspect

ratio between the initial water depth and the lock length, and the

initial excess density driving the motion. For the ﬁrst time diﬀer-

ent methodologies are used in order to evaluate entrainment and

mixing during the slumping phase: two entrainment parameters,

the variation of the dense current fractional area and an energy

budget method (Winters et al. 1995). Part of the cases herein

investigated develop the self-similar regime, allowing the study

of mixing also during this second phase. This work contributes

to a better entrainment parametrization and to a better know-

ledge of mixing in unsteady gravity currents. The novel results

of this work will be helpful in the implementation of simpliﬁed

models which are not able to directly reproduce entrainment and

mixing processes, such as shallow water models. The LES of the

present study are also validated against laboratory experiments,

which have been carried out in conditions as similar as possible

to the corresponding numerical simulation.

The paper is organized as follows. In Section 2the labora-

tory experiments are presented together with the experimental

set-up. In Section 3the numerical model and the simulations

are described. Section 4shows a preliminary analysis of the

numerical results and a comparison with the experimental data.

In Section 5the entrainment is studied by the deﬁnition of two

entrainment parameters. In Section 6the variation of the current

fractional area is discussed depending on the threshold value

chosen to deﬁne the interface between the ambient and the dense

Journal of Hydraulic Research Vol. 54, No. 5 (2016) Entrainment and mixing in unsteady gravity currents 543

ﬂuids. In Section 7mixing is analysed using the energy budget

method. Finally, conclusions are given in Section 8.

2 Laboratory experiments

The laboratory experiments are performed at the Hydraulics

Laboratory of the University “Roma Tre,” using a Plexiglas tank

3 m long, 0.2 m wide and 0.3 m deep (Fig. 1). The size of the

tank is similar to those commonly used in lock-exchange experi-

ments (e.g. Dai 2013; Fragoso et al. 2013). A removable gate

is placed at a distance x0from the left wall of the tank divid-

ing the tank into two diﬀerent volumes. The volume on the

right-hand side of the gate is ﬁlled with fresh water of den-

sity ρ0, while the volume on the left-hand side is ﬁlled with

salty water of density ρ1>ρ

0. Both volumes are ﬁlled up to the

same height h0. A full-depth gravity current is thus generated

with h0/H=1, where H=0.2 m is the depth of the ambient

ﬂuid and h0=0.2 m is the depth of the ﬂuid in the lock. The

aspect ratio r=h0/x0of the initial volume of the dense ﬂuid in

the lock is varied between 0.5 and 2 in order to investigate the

inﬂuence of ron mixing. A controlled quantity of dye (E171,

titanium dioxide) is added to the salty water to make visible the

dense ﬂuid. Densities ρ1and ρ0are measured by a pycnometer.

After the gate removal, the dense column of water in the lock

collapses and a gravity current develops.

The experiments are recorded by a CCD (charged coupled

device) video camera with a resolution of 768 ×576 pixels

and an acquisition frequency of 25 Hz. The recorded images

are cropped in order to focus on the region of the tank. Black

and white images are then converted in matrices of grey lev-

els with values between 0 (black) and 255 (white). A threshold

method (Adduce et al. 2012) is applied to detect the interface

between the dense and the ambient ﬂuids. Nine experiments

(EXP1 ÷EXP9) were performed, details of which are given in

Table 1. The density ﬁelds for EXP1, EXP2, EXP3, EXP4 and

EXP7 were evaluated using dye as a tracer. For each pixel of

the image, a relation between the amount of the uniformly dis-

tributed dye in the tank and the grey scale values was obtained

by acquiring, at the end of each experiment, images character-

ized by a known concentration of dye (as in Nogueira, Adduce,

Alves, & Franca 2013b).

Figure 1 Sketch of a Plexiglas tank for lock-exchange experiments

Table 1 Experimental conditions

Run

g

0

(m s−2)x0(m) r

Ubulk

(m s−1)RR

bFslump

EXP1 0.15 0.1 2 0.048 4840 34373 0.39

EXP2 0.29 0.1 2 0.071 7108 48345 0.40

EXP3 0.59 0.1 2 0.105 10522 68555 0.45

EXP4 0.15 0.2 1 0.066 6629 34236 0.44

EXP5 0.29 0.2 1 0.099 9859 48265 0.46

EXP6 0.59 0.2 1 0.142 14228 68564 0.46

EXP7 0.15 0.4 0.5 0.077 7738 34403 0.45

EXP8 0.29 0.4 0.5 0.113 11265 48275 0.47

EXP9 0.59 0.4 0.5 0.160 16010 68560 0.47

Two experimental parameters are varied: rand the initial

excess density , deﬁned as:

=ρ1−ρ0

ρ0

(1)

The initial reduced gravity g

0is deﬁned as g

0=g. The ﬂuid

densities ρ1,ρ0are chosen in order to obtain three diﬀerent val-

ues of the excess density (=0.015, =0.030 and =0.060),

and consequently the values of g

0are 0.15 m s−2,0.29ms

−2and

0.59 m s−2. For each experiment a bulk Reynolds number Rand

a buoyancy Reynolds number Rbare evaluated as:

R=Ubulkh0/2

ν(2)

Rb=ubH

ν(3)

where νis the kinematic viscosity and Ubulk is the mean velocity

of the current calculated as the ratio between the total distance

travelled by the current and the duration of the experiment. ubis

the buoyancy velocity, deﬁned as:

ub=g

0H(4)

A Froude number characterizing the slumping phase, Fslump,is

evaluated using as velocity scale the bulk velocity at the end

of the slumping regime Ubsl (i.e. the ratio between the distance

travelled by the current in the slumping phase xsand the related

slumping time ts):

Fslump =Ubsl

g

0h0

(5)

3 Numerical set-up

We use an improved parallel version of the LES model

employed in Armenio and Sarkar (2002) for the investigation

of stable stratiﬁed wall bounded ﬂows. This model, named

LES-COAST, is able to reproduce real and laboratory scale

environmental water dynamics (Petronio, Roman, Nasello, &

544 L. Ottolenghi et al. Journal of Hydraulic Research Vol. 54, No. 5 (2016)

Armenio 2013). Over the years, the model has been applied

to simulate a large variety of ﬂuid-dynamics phenomena, even

characterized by rapid transition to turbulence and relaminariza-

tion (see, among others, Roman, Stipcich, Armenio, Inghilesi, &

Corsini 2010; Salon, Armenio, & Crise 2007). Here, the model

is used to investigate the dynamics of lock-release gravity cur-

rents. The model solves the ﬁltered Navier–Stokes equations

under the Boussinesq approximation:

∂¯uj

∂xj

=0(6)

∂¯ui

∂t+∂¯uj¯ui

∂xj

=−1

ρ0

∂¯p

∂xi

+ν∂2¯ui

∂xj∂xj

−ρ

ρ0

gδi2−∂τij

∂xj

(7)

∂¯s

∂t+∂¯uj¯s

∂xj

=ks

∂2¯s

∂xj∂xj

−∂lj

∂xj

(8)

where uidenotes the velocity component in the xi-direction. The

overbar denotes the ﬁltering operation consisting in the sepa-

ration between the small and dissipative scales of the motion

from the resolved ones (Pope 2000). The parameters νand ks

in Eqs (7) and (8) are the kinematic viscosity and the molec-

ular salt diﬀusivity, respectively; the gravitational acceleration

acting along x2or ydirection is g; the hydrodynamic pressure

and the salinity are pand s, respectively; the variation of den-

sity with respect to the reference value ρ0(corresponding to the

reference salinity s0)isρ. The density ρ(x,y,z,t)is related to

the variation of salinity as shown in the state Eq. (9):

ρ=ρ0[1 +β(s−s0)](9)

where βis the salinity cubic contraction coeﬃcient. In Eq. (9)

the dependence of ρfrom temperature is omitted because in this

study the ﬁeld is isothermal.

In LES, through the ﬁltering operation, the large, anisotropic

and energy carrying scales of turbulence are directly resolved,

while smaller scales are modelled. The unresolved scales of

motion are represented by the subgrid scale (SGS) momentum

and salinity ﬂuxes τij and ljof Eqs (7) and (8) respectively.

The SGS momentum and salinity ﬂuxes are modelled using

the dynamic Smagorinsky eddy viscosity model. The model is

described in detail in Armenio and Sarkar (2002). In order to

solve the SGS ﬂuxes, two variables are introduced in the model,

which are here calculated dynamically using the Lagrangian

methodology of Meneveau, Lund, and Cabot (1996). The

governing equations are integrated using the semi-implicit,

fractional-step method of Zang, Street, and Koseﬀ (1994).

The algorithm resolves explicitly the time advancement of the

advective terms through the second-order Adams–Bashforth

technique. The implicit Crank–Nicolson scheme is used to cal-

culate the diﬀusive terms, while a second-order centred scheme

discretizes the spatial derivatives. A SOR algorithm is used for

the resolution of the pressure equation together with a multigrid

technique to enhance the convergence rate.

In the numerical simulations physical parameters consistent

with those of the experiments are used. The numerical domain

has dimensions Lx=4.096 m, Ly=0.2 m and Lz=0.2 m. The

length-to-height ratio is equal to 20.48. The numerical grid is

made of 2048 ×128 ×64 cells respectively in the streamwise,

vertical and spanwise direction. The grid spacings along the

streamwise and spanwise directions are respectively 0.01 Hand

0.016 H. Grid cells are non-uniform along the vertical direction

with spacing ranging from 0.01 Hat the top to 0.002 Hat the

bottom where a transitional boundary layer develops. The spa-

tial resolution is similar to wall-resolving LES already used for

the analysis of gravity currents (Tokyay et al. 2011). The a pos-

teriori analysis of the numerical results assures that values of

x+,y+and z+are of the order of 50, 1 and 20, respec-

tively. These are the values of reference needed to avoid the use

of a wall function, indicating that the viscous sublayer is directly

solved by the LES model. The Schmidt number (Sc), deﬁned as

the ratio between the kinematic viscosity and the molecular dif-

fusivity, is set to Sc =600 (the reference value for salt water).

The time-step of the simulations is adjusted by the model in

order to keep a constant Courant number =0.6.

A shear-free boundary condition is employed at the top

boundary to mimic free-surface experimental condition, as

shown by Liu and Jiang (2013). Periodic conditions are

employed along the spanwise direction so that the inﬂuence of

lateral walls is not considered and large relative width ﬂows are

simulated. No-slip conditions are set at the remaining bound-

aries. The ﬂow ﬁeld is initialized with the ﬂuid at rest every-

where. A spatial distribution of the scalar is imposed at t=0

with a discontinuity in correspondence of the plane x=x0:the

salinity values are ﬁxed to obtain the initial density value of ρ1

on the left of x0, whereas the density is equal to ρ0elsewhere.

Zero ﬂux of the scalar is imposed at all boundaries.

Nine diﬀerent simulations are performed varying both the

initial density of the heavy ﬂuid ρ1and the aspect ratio rof

the ﬂuid in the lock. The ambient ﬂuid has a density ρ0=

1000 kg m−3and the initial water depths H=h0=0.2 m in all

the cases investigated. Table 2summarizes the parameters of the

numerical simulations.

Table 2 Parameters for numerical simulations

Run

g

0

(m s−2)x0(m) r

Ubulk

(m s−1)RR

bFslump

RUN1 0.15 0.100 2 0.049 4891 34310 0.42

RUN2 0.29 0.100 2 0.075 7500 48522 0.43

RUN3 0.59 0.100 2 0.105 10486 68621 0.43

RUN4 0.15 0.200 1 0.058 5809 34310 0.43

RUN5 0.29 0.200 1 0.091 9082 48522 0.43

RUN6 0.59 0.200 1 0.129 12859 68621 0.44

RUN7 0.15 0.400 0.5 0.075 7498 34310 0.44

RUN8 0.29 0.400 0.5 0.106 10569 48522 0.44

RUN9 0.59 0.400 0.5 0.152 15223 68621 0.44

Journal of Hydraulic Research Vol. 54, No. 5 (2016) Entrainment and mixing in unsteady gravity currents 545

4 Numerical results and comparison with the experiments

In order to assess the reliability of the numerical model to simu-

late gravity currents, the results of the numerical simulations are

compared to those obtained in the laboratory experiments. The

numerical dimensionless density ﬁeld, ρ∗, is deﬁned as:

ρ∗(x,y,z,t)=ρ(x,y,z,t)−ρ0

ρ1−ρ0

(10)

Here, statistics are accumulated averaging along the spanwise

direction of homogeneity. This operation is denoted with the

symbol . Diﬀerent density thresholds can be chosen to deﬁne

the interface between the gravity current and the ambient ﬂuid.

The aim of the present study is the investigation of entrain-

ment and mixing in lock-release gravity currents, in order to

give a contribution to the entrainment parametrization for sim-

pliﬁed models which are not able to solve the small scale mixing

processes, i.e. shallow water models. These models need to

parametrize the entrainment by a source or a sink term in the

continuity equations for the two layers (see Adduce et al. 2012;

Lombardi et al. 2015;Rossetal.2006). In two-layer shallow

water models the detrainment of dense ﬂuid is not accounted for,

then the ambient ﬂuid is assumed to not vary its density. Conse-

quently, in the present study the iso-density level ρ∗=0.02

is chosen to deﬁne the interface between the two ﬂuids, in

order to consider as dense current the part of the ﬂuid that is

not purely ambient ﬂuid, as in the simpliﬁed two-layer shallow

water approach. The inﬂuence of the iso-density threshold cho-

sen to deﬁne the interface between the gravity current and the

ambient ﬂuid is discussed in Section 6.

A comparison between laboratory measurements and numer-

ical simulations in terms of the advancement of the front is

shown in Fig. 2, where the streamwise coordinate of the front

position xfversus time is plotted. xfis deﬁned as the location

of the foremost point of the nose of the gravity current along

the x-axis. In the laboratory experiments, the foremost point

of the nose of the gravity current is found to be located at a

dimensionless distance of about 0.02 Hfrom the bed. Then, it is

evaluated for both the laboratory experiments and the numerical

simulations at the distance of 0.02 Hfrom the bottom bound-

ary. The dependence of xfon the spanwise direction is found

to be negligible. Henceforth, where not otherwise indicated,

spanwise averaged values of density ρand velocities uiare

considered.

The dimensionless front position is deﬁned as:

x∗

f(T∗)=xf(t)−x0

x0

(11)

where the dimensionless time T∗is deﬁned as:

T∗=t∼ub

x0

(12)

The quantitative comparison between the numerical (xfnum) and

the experimental (xfex) front positions is shown in Fig. 2.In

all cases, the simulations reproduce well the time evolution of

the front position. In order to give an estimate of the overall

accuracy, a mean error Err is deﬁned as:

Err =mean

xfex(t)−xfnum (t)

xfex(t)

(13)

The value of this quantity is 1.9, 2.7, 3.8, 4.3, 6.3, 5.6, 4.7,

6.1 and 4.9% for RUN1, RUN2, RUN3, RUN4, RUN5, RUN6,

RUN7, RUN8 and RUN9, respectively, suggesting a reason-

able agreement between the numerical and the experimental

positions of the fronts. In Fig. 2the numerical and the exper-

imental xf(t)are shown for the three diﬀerent values of rtested:

Fig. 2a reports the comparison of xf(t)for the cases in which

the conﬁguration of the ﬂuid domain in the lock at t=0is

vertically-oriented (r=2); in Fig. 2b the front positions of the

gravity currents corresponding to a square-shaped initial lock

ﬂuid domain are displayed (r=1); ﬁnally, in Fig. 2cxf(t)is

shown for the gravity currents with the horizontally-oriented

initial lock ﬂuid domain (r=0.5). As expected, gravity cur-

rents with high g

0are faster than the currents with low g

0.The

velocity of gravity currents is higher as rdecreases. Finally,

(a) (b) (c)

Figure 2 Numerical and experimental front positions versus time: (a) r=2; (b) r=1; (c) r=0.5. Square markers and dash-dot lines indicate the

g

0=0.59 m s−2cases; g

0=0.29 m s−2cases are shown by dashed lines and circles; triangle markers and solid lines are for the g

0=0.15 m s−2

cases

546 L. Ottolenghi et al. Journal of Hydraulic Research Vol. 54, No. 5 (2016)

the curves of xf(t)in Fig. 2are more bent in the vertically-

oriented domains than in the horizontally-oriented domains. In

fact, the front positions in Fig. 2c are represented by straight

lines, indicating a constant front velocity and the development

of the slumping phase only.

In order to analyse the phases of the gravity currents, a

log–log plot of x∗

fversus T∗is used, as in Rottman and

Simpson (1983). Since Err is small and the behaviour is simi-

lar in all the cases analysed, selected runs are plotted in Fig. 3

as examples. In Fig. 3a the cases in which the initial ﬂuid

domain is vertically-oriented are presented in order to show the

eﬀect of the variation of g

0on the advancement of the dimen-

sionless front position: x∗

f(T∗)curves collapse on a single one

independent of g

0, as expected. The eﬀect of the variation of

rfor ﬁxed values of g

0is shown in Fig. 3b, where RUN2,

RUN5 and RUN8 are plotted. In all the experiments and sim-

ulations, when the gate dividing the heavy and the light ﬂuids

is removed, the column of dense ﬂuid collapses over the bot-

tom of the tank. A gravity current starts moving downstream,

while the ambient ﬂuid above moves backward, ﬁlling the vol-

ume left by the dense ﬂuid. This backward wave is then reﬂected

by the left wall of the tank and a bore starts moving along the

same direction as the heavy ﬂuid. The velocity of the bore is

higher than that of the current, thus the bore overtakes the nose

of the current after a while. The dashed lines with slope =1

in Fig. 3mark the slumping phase, during which the front of

the current moves at a constant velocity. In this ﬁrst phase, the

bore and the current are travelling downstream and x∗

fincreases

linearly in time. The slumping phase is visible in all the simu-

lated cases, and it is well approximated by the linear function

with slope equal to 1. Once the bore reaches the nose of the

current and overtakes it, a transition to the self-similar phase

takes place. In the self-similar phase the dynamics of the cur-

rent are governed by the balance between inertial and buoyancy

forces. During this second phase the current slows down and x∗

f

evolves according to the theoretical power law of t2/3. Although

this transition occurs gradually, several authors deﬁned a crite-

rion in order to mark the passage from one regime to the other

(Nogueira, Adduce, Alves, & Franca 2013a; Ooi et al. 2009;

Shin, Dalziel, & Linden 2004). In the present work the approach

ofOoietal.(2009) is used to deﬁne the slumping time ts,

and the transition from the slumping to the inertial regime was

observed to occur after the gravity current has ﬂown for about

7.7 lock-lengths. All the gravity currents slow down during the

self-similar phase both in the simulations and in the experi-

ments, and the theoretical trend is reported in Fig. 3with the

solid thin lines with a slope =2/3. The cases corresponding to

vertically-oriented initial lock ﬂuid domains ﬂow for 29 lock-

lengths, indicating that both the slumping and the self-similar

phases are simulated. The gravity currents with the square-

shaped initial lock ﬂuid domain develop for 14 lock-lengths,

indicating that the dense ﬂow is simulated along the whole

slumping regime and also in the ﬁrst part of the self-similar

one. In the cases with a horizontally-oriented initial lock ﬂuid

domain the gravity currents ﬂow within the slumping regime,

because the dense current reaches the end of the tank after it has

travelled for 6.5 lock-lengths. Thus, observing the evolution in

time of the dimensionless front positions, the cases with diﬀer-

ent rjust exhibit diﬀerent dimensionless durations, because at

the end of the experiments diﬀerent ﬂow regimes are reached

(Fig. 3b). Regarding the evolution in time of the dimensionless

front position, it clearly appears that it is unaﬀected by either g

0

or r, both in the slumping phase and at the beginning of the self-

similar one (Fig. 3a and b, respectively), since x∗

f(T∗)collapses

onto the same curve for all cases examined.

A qualitative comparison between the density ﬁelds of the

gravity currents obtained by the analysis of the laboratory

experiments (EXP1, EXP2, EXP3, EXP4 and EXP7) and the

corresponding numerical simulations is also performed. As an

example, Fig. 4shows the contour maps of ρ∗taken during

RUN1 and the concentration ﬁelds gained from the experi-

mental pictures captured during EXP1 at the same times. The

behaviour of the current during the numerical simulation is

shown in Fig. 4b, d, f and h. The corresponding images from

the laboratory experiment are in Fig. 4a, c, e and g. The main

features of the numerical gravity current are rather similar to the

(a) (b)

Figure 3 Dimensionless front position versus dimensionless time. (a) Eﬀect of the variation of g

0for r=2: RUN1, RUN2, RUN3; (b) eﬀect of the

variation of rfor g

0=0.29 m s−2: RUN2, RUN5 and RUN8

Journal of Hydraulic Research Vol. 54, No. 5 (2016) Entrainment and mixing in unsteady gravity currents 547

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 4 Experimental density ﬁelds of EXP1 (left) and numerical density ﬁelds of RUN1 (right) at diﬀerent times (g

0=0.15 m s−2,r=2):

(a) and (b) T∗=13.6; (c) and (d) T∗=35.7; (e) and (f) T∗=65.7; (g) and (h) T∗=91.3. Iso-density levels correspond to ρ∗=0.02, ρ∗=0.05,

ρ∗=0.08, ρ∗=0.1, ρ∗=0.2 and ρ∗=0.5

experimental ones. The bulk structure of the dense current along

the x-axis is reasonably reproduced by the numerical model. In

both cases the height of the head of the current decreases as

time increases. The tail region increases in length and becomes

thinner at the end of the simulation. Diﬀerences in the cur-

rent’s behaviour can be observed in the area where the transition

between the body and the tail regions develops: in the numeri-

cal simulations the end of the body is visible as a minimum

of the iso-density level dividing the current from the ambient

ﬂuid, while this transition is not evident in the laboratory experi-

ments. Nevertheless, a fair agreement between the numerical

and the experimental density ﬁelds was found, conﬁrming that

the numerical model is suﬃciently able to simulate the dyna-

mics of gravity currents. Furthermore, the increase in volume

of the gravity currents during their propagation is reasonably

reproduced by the numerical simulations, as will be shown in

Section 5. Diﬀerences between the numerical results and labo-

ratory experiments are inevitably associated to the eﬀects of the

gate removal, as well as to the boundary conditions imposed

in the model. They are similar but not the same as in the

experimental situations.

4.1 Density ﬁelds

The features of the density ﬁeld ρ∗are shown in Fig. 4for

RUN1. As the gravity current moves along the tank entraining

ambient ﬂuid and mixing with it, the ﬂuid density decreases in

time (Fig. 4b, d, f and h). At T∗=13.6 the interface of the cur-

rent is characterized by rounded bulges, under which a billow

starts to develop (Fig. 4b). Cases with higher values of g

0and

diﬀerent rdiﬀer primarily in the thickness of the gravity current,

in the front velocity, and in the distribution of the iso-density

(a)

(b)

(c)

(d)

Figure 5 ρ∗of RUN9 (g

0=0.59 m s−2and r=0.5) at diﬀerent

times: (a) T∗=3.9; (b) T∗=7.7; (c) T∗=12.2; (d) T∗=14.5. Iso–

density levels correspond to ρ∗=0.02, ρ∗=0.1, ρ∗=0.2 and

ρ∗=0.5

lines deﬁning the structure of the density ﬁeld. As an example,

Fig. 5shows the evolution of ρ∗for RUN9 at diﬀerent times.

From the comparison of the darkest areas in Figs 4and 5,it

appears that in the r=0.5 case the core of the current remains

relatively unmixed. This aspect was also observed in the experi-

ments of Hacker et al. (1996). In all the other cases herein

considered, the evolution in time of ρ∗shows a behaviour

intermediate between RUN1 and RUN9.

In order to study the three-dimensionality of the ﬂow, 3D

dimensionless density ﬁelds are here analysed. The iso-surface

corresponding to ρ∗=0.02 is shown in Fig. 6for RUN1 as

an example (r=2 and g

0=0.15 m s−2); in this case the ﬂow

548 L. Ottolenghi et al. Journal of Hydraulic Research Vol. 54, No. 5 (2016)

(a)

(b)

(c)

Figure 6 Three-dimensional density iso-surface, ρ∗=0.02, of the gravity current for RUN1 at: (a) T∗=19; (b) T∗=34; (c) T∗=88

develops both the slumping and the self-similar regimes. Dur-

ing the slumping phase (Fig. 6a) the ﬂow is almost 2D and

Kelvin–Helmholtz billows extend along the spanwise direction

of the ﬂow. The behaviour of the ﬂow becomes more complex

after ts(Fig. 6b and c): the current slows down, billows start to

lose their coherence and the ﬂow structure becomes more three-

dimensional (in agreement with Cantero et al. 2007,2008).

Other typical structures characterizing the dynamics of a gravity

current are the lobe-and-cleft instabilities. These structures are

visible in the region behind the nose of the current during all

the simulations and characterize the shape of the gravity current

along the spanwise direction. They become more evident with

the decay of the Kelvin–Helmholtz instabilities (Fig. 6c) and

inﬂuence the behaviour of the interface during the self-similar

phase through an up-stream propagation of three-dimensional

disturbances. A developed 3D ﬂow, with turbulent perturba-

tions aﬀecting the whole body and tail of the gravity current,

is clearly visible in Fig. 6c. A transition in behaviour is evi-

dent at x/H=6 (Fig. 6c), due to the decay of turbulence in

the tail region of the dense current (Cantero et al. 2007,2008;

Ooi et al. 2009). The increase of g

0aﬀects the dynamics of

the ﬂow enhancing turbulence, and thus density iso-surfaces

are more complex in RUN2 and RUN3 than in RUN1, partic-

ularly at the end of the simulations (not shown). The decrease

of rincreases the thickness and the velocity of the gravity cur-

rents and, consequently, the turbulence is enhanced. A more

pronounced three-dimensionality of the ﬂow can be detected

for low r, although the limited dimensions of the domain leads

to the occurrence of the slumping phase only (not shown).

In particular, the density iso-surface is characterized by small

scale turbulent structures and no relevant coherent Kelvin–

Helmholtz billows are visible. This may be ascribed to the

diﬀerent behaviour of the billows characterizing the r≤1 cases.

They constitute a large population of small structures located

under the iso-surface ρ∗=0.02, in the internal part of the cur-

rent. This causes the strongly chaotic and three-dimensional

behaviour of the interface dividing the dense and the ambient

ﬂuids.

5 Entrainment in gravity currents

When a gravity current ﬂows, the initial potential energy is

converted into kinetic energy and mixing occurs between the

current and the ambient ﬂuid. Along its path, the gravity current

entrains fresh water and increases its volume. The entrainment

velocity can be modelled as a fresh water discharge per unit

area crossing the interface between the two ﬂuids, which causes

an increase in volume of the gravity current (as in Adduce

et al. 2012; Cenedese & Adduce 2008; Nogueira, Adduce,

Alves, & Franca 2014). The entrainment can be described by

the time-variation of the volume of the gravity current. If at

t0=0 s the initial volume of the gravity current is V0,atasuc-

cessive time ti=t0+tithe dense current has a larger volume

Vi=V0+Vi, with Vievaluable by:

Vi=(Ai−A0)d(14)

Journal of Hydraulic Research Vol. 54, No. 5 (2016) Entrainment and mixing in unsteady gravity currents 549

where dis the spanwise dimension of the channel, Aiis the area

under the iso-density level ρ∗=0.02 in the vertical x–yplane

at tiand A0is the initial area of the dense current at t0=0. A

bulk entrainment discharge can be calculated at each time by:

Qei =Vi

ti

(15)

and a bulk entrainment velocity, Wei, crossing the interface Si

separating the two ﬂuids with diﬀerent density, is deﬁned as:

Wei =Qei

Si

(16)

where the interface is scaled by Si=dx

ﬁ. The bulk entrainment

parameter Eiis a dimensionless number obtained dividing the

bulk entrainment velocity by a velocity scale given by the dif-

ference V1−V2(Turner 1973), where V1is the bulk velocity

of the density current and V2is the bulk velocity of the ambi-

ent ﬂuid. Then the bulk entrainment parameter, Ei_bulk, can be

deﬁned as:

Ei_bulk =Wei

2Ui_bulk

(17)

where Ui_bulk is the bulk velocity of the current at a certain time

ti. An instantaneous entrainment parameter Ei_inst is here deﬁned

by evaluating the increase in volume of the current at a certain

time ti, with respect to the volume at the previous time ti−1. Then

relations (15)–(17) become:

Qei_inst =(Ai−Ai−1)d

(ti−ti−1)(18)

Wei_inst =Qei_inst

1

2(Si+Si−1)(19)

Ei_inst =Wei_inst

(Ui_bulk +Ui−1_bulk)(20)

Changes in volume of the gravity current, indicating the occur-

rence of instantaneous mixing, are directly related to the abrupt

variation of the instantaneous entrainment parameter. The evo-

lution of the bulk and the instantaneous entrainment parameters

evaluated for all the simulations are shown in Fig. 7a–i. The

bulk entrainment parameter for an unsteady gravity current

decreases as the front position advances, as conﬁrmed also by

the laboratory measurements of Cenedese and Adduce (2008)

for steady gravity currents. In addition, Ebulk reaches a ﬁnal

value of the order of 10−2, in agreement with the previous labo-

ratory evaluations for both steady and unsteady gravity currents

of Adduce et al. (2012), Cenedese and Adduce (2008,2010),

and Nogueira et al. (2014). The instantaneous entrainment

parameter has a more complex behaviour, due to the irregu-

larity of mixing occurring in a lock-release gravity current. At

the beginning of the simulations, the column of dense ﬂuid

collapses and a turbulent ﬂow is generated: the ﬁrst point of

Einst represents the large and sudden increase of volume from

t=t0(initial lock volume at rest) to t=t1(volume of the dense

current at the ﬁrst time of the simulation). Once the gravity

current is generated, a reduction of the entrainment velocity

occurs and Einst decreases in all the simulations. In the runs with

avertically-oriented initial lock ﬂuid domain (RUN1, RUN2

and RUN3), two major peaks of the instantaneous entrain-

ment parameter can be observed at (xf−x0)/H1 and when

(xf−x0)/H4 (Fig. 7a, d and g). In the runs correspond-

ing to square-shaped initial lock ﬂuid domains (RUN4, RUN5

and RUN6) the peaks decrease in intensity and Einst becomes

more irregular (Fig. 7b, e and h). In particular, a second peak

is not clearly discernible, and irregularities are visible up to

(xf−x0)/H6. A smoother proﬁle of Einst is reached after

(xf−x0)/H7.7, when the transition to the self-similar phase

occurs. The gravity currents with a horizontally-oriented ini-

tial lock ﬂuid domain, RUN7, RUN8 and RUN9, are shown in

Fig. 7c, f and i, respectively. The ﬁrst peak in each case is still

clearly visible, although reduced if compared with those with

higher r. Irregular features are observed during all the rest of the

simulations, and secondary peaks are hardly discernible.

In order to better understand the processes associated with

the major peaks observed in Einst,ther=2 cases are further

analysed and discussed. Three dots (denoted by A, B, C and

D, E, F) are depicted for each peak in Fig. 7a: the instant at

which the peak veriﬁes (B and E), the time before (A and D)

and after the occurrence of the peak (C and F). The entrainment

processes occurring when the two peaks take place can be also

investigated by means of the contour of ρ∗shown in Figs 8a–f

for RUN1 where the times correspond to the occurrence of the

points A, B, C, D, E and F in Fig. 7a.

At the beginning of the simulation, positive vorticity char-

acterizes the region at the interface between the current and

the ambient ﬂuid and Kelvin–Helmholtz billows start to form

(Fig. 8a). These billows aﬀect the features of the gravity cur-

rent, curving the trajectories of the dense ﬂuid particles, and

homogenizing the density ﬁeld inside the body of the current.

Later on, these structures increase in size and in intensity, and

their eﬀects are no longer conﬁned within the current region

only. When the billow grows and reaches the densest internal

part of the gravity current, the ﬂuid of larger density is involved

in the development of the billow itself. Part of the dense ﬂuid

is lifted up into the ambient ﬂuid and a streak of heavy ﬂuid is

bent upstream. When the curved path of these dense streaks is

completed at the rear of the current, light ﬂuid is trapped into

the gravity current (Fig. 8b). At these instants the maximum in

Einst is observed (Fig. 7a, point B). Successively, while the gra-

vity current continues to ﬂow downstream, billows decrease in

size and move down inside the body of the dense ﬂow, leading

to a smoother interface of the gravity current (Fig. 8c). During

later times, Kelvin–Helmholtz instabilities continue to develop,

aﬀecting Einst, whose rapid changes are visible in Fig. 7a.

At the end of the slumping phase another peak is found in all

the simulations with r=2 (Fig. 7a, point E). Again, big billows

lift up the dense ﬂuid into the lighter one (Fig. 8d) and, during its

550 L. Ottolenghi et al. Journal of Hydraulic Research Vol. 54, No. 5 (2016)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 7 Bulk entrainment parameter (black line) and instantaneous entrainment parameter (grey line) versus dimensionless front position:

(a) RUN1; (b) RUN4; (c) RUN7; (d) RUN2; (e) RUN5; (f) RUN8; (g) RUN3; (h) RUN6; (i) RUN9. Dots labelled as A–F are used as references for

the analysis of the density ﬁelds of Fig. 8(g

0is in m s−2)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8 ρ∗at diﬀerent times for RUN1. (a) T∗=3.7 and x∗

f=1.6; (b) T∗=5.1 and x∗

f=2.1; (c) T∗=7.0 and x∗

f=3; (d) T∗=18.0

and x∗

f=7.6; (e) T∗=19.4 and x∗

f=8.2; (f) T∗=19.9 and x∗

f=8.3. The density ﬁelds shown in panels (a)–(f) correspond to the points A–F,

respectively, of Fig. 7a

development, streaks of light ﬂuid are trapped inside the current

(Fig. 8e). White streaks of light ﬂuid trapped inside the current

are clearly visible in Fig. 8e at about x/H=3.5. This trapped

ambient ﬂuid starts to mix with the dense part of the current

at later times (Fig. 8f). During the rest of the simulations the

currents continue to mix with the ambient ﬂuid with a slower

rate as shown by the entrainment parameters of Fig. 7a, d and g.

The main action of a Kelvin–Helmholtz billow relates to

forcing a curved trajectory for the ﬂuid particles inside the body

of the current. Thus, Kelvin–Helmholtz instabilities produce

Journal of Hydraulic Research Vol. 54, No. 5 (2016) Entrainment and mixing in unsteady gravity currents 551

mixing mainly in the body of the current, making its interior

more homogeneous. On the other hand, these billows also catch

ambient ﬂuid, trapping it into the dense current, and producing

an eﬃcient mixing after their collapse. The same analysis car-

ried out for RUN2 and RUN3 (not shown here) gives similar

results. In the cases with r≤1 the Kelvin–Helmholtz insta-

bilities are related to billows less prone to capture parcels of

ambient ﬂuid but are more stable and persistent than in the r>1

cases. The billows are in fact conﬁned into the body of the

gravity current and they remain more compact.

The value Eof the entrainment parameter Ebulk at the end

of the simulation depends on the Reynolds number, the Froude

number and on the length of the path followed by the gravity

current (Cenedese & Adduce 2008,2010). The dense currents

characterized by r=0.5 ﬂow at most for 2.6 m in the present

experimental apparatus. Thus, in order to compare the diﬀerent

cases here analysed, the Evalues are evaluated as Ebulk at (xf−

x0)=2.6 m. The bulk Froude number Fis here deﬁned as:

F=Ubulk

g

m

h0

2

(21)

where

g

m=g

0+g

end

2(22)

and g

end is calculated with the mean density of the current at the

end of the simulations.

Figure 9a and b show the values of Eversus Fand ver-

sus R, respectively, for all the simulations and for some of the

experiments performed. The order of magnitude of E∼10−2,

obtained by the present numerical simulations and the exper-

iments, is in agreement with previous laboratory studies of

lock-release gravity currents (Adduce et al. 2012; Nogueira

et al. 2014). Moreover, the present experimental evaluations of

Eare in fair agreement with the numerical results. For a ﬁxed

value of g

0,ifrdecreases the Froude and the Reynolds numbers

increase and, consequently, the entrainment is enhanced. In fact,

for cases with vertical orientation of the initial lock ﬂuid domain

F0.6, 4500 <R<10500 and E0.0011. The cases with a

square-shaped initial lock ﬂuid domain are in the central part

of both Fig. 9a and b, with F0.7, 5500 <R<13000 and

E0.0014. Finally, for cases with horizontal orientation of the

initial lock ﬂuid domain the entrainment is larger than the cases

with r≥1(E0.00155). Thus raﬀects the amount of ambient

ﬂuid entrained by the dense current. Also g

0slightly aﬀects E:

for a given r, in most of the simulations if g

0increases, the same

occurs for Fand R, and, consequently Eincreases. From the

analysis of Ebulk,Einst and E, it is possible to argue that in the

r=2 cases the peaks of Einst are most pronounced, since they

are associated to the trapping of ambient ﬂuid during the devel-

opment of Kelvin–Helmholtz billows. Nevertheless, the global

amount of fresh ﬂuid entrained by the current is larger in the

r=0.5 cases, as shown in Fig. 9.

In Fig. 10 Eversus Fis plotted together with some entrain-

ment parametrizations for both steady and unsteady gravity

Figure 10 Eversus F: present results (stars) and entrainment

parametrizations for both steady and unsteady gravity currents:

Turner (1986) (dashed line); Parker et al. (1987) (grey line with dots);

Ross et al. (2006) (solid line with circles); Cenedese and Adduce (2010)

depending on R(black solid lines); Adduce et al. (2012) (solid line with

crosses)

(a) (b)

Figure 9 (a) Eversus F;(b)Eversus R. Empty, bold and black-ﬁlled markers indicate r=2, r=1andr=0.5, respectively. Squares are

for g

0=0.15 m s−2cases, triangles indicate g

0=0.29 m s−2cases and circles mark g

0=0.59 m s−2cases. Grey markers indicate entrainment

evaluations from laboratory measurements

552 L. Ottolenghi et al. Journal of Hydraulic Research Vol. 54, No. 5 (2016)

currents. The entrainment parametrizations of Turner (1986),

Parker et al. (1987), and Cenedese and Adduce (2010) are based

on the laboratory evaluation of the entrainment in steady gravity

currents, i.e. currents fed by a constant discharge of dense water.

The relations of Ross et al. (2006) and Adduce et al. (2012)

are proposed and used to parametrize the entrainment in two-

layer shallow water models simulating lock-release gravity

currents, i.e. unsteady gravity currents. The present entrain-

ment evaluations refer to unsteady gravity currents produced

by the lock-exchange technique. These currents are characteri-

zed by low values of F(F<1) and large values of R.The

relation of Turner (1986) does not predict any entrainment for

subcritical gravity currents (F<1), while some recent stud-

ies (Adduce et al. 2012; Cenedese and Adduce 2008; Fragoso

et al. 2013; Hacker et al. 1996; Nogueira et al. 2014) detected

the occurrence of entrainment for gravity currents with F<1.

The entrainment parametrization of Parker et al. (1987) accounts

for subcritical mixing. Further, the parametrization presented

in Cenedese and Adduce (2010) takes into account also the

dependence of the entrainment on R. The value of Epre-

dicted by the parametrizations for steady gravity currents of

Parker et al. (1987) and Cenedese and Adduce (2010)forF<1

ranges between 1 ×10−4<E<2×10−3and underpredicts

the entrainment evaluations of the present study on unsteady

gravity currents. The present LES and laboratory entrainment

evaluations are in agreement with the entrainment parameters

observed for subcritical lock-exchange gravity currents (Adduce

et al. 2012; Nogueira et al. 2014). On the other hand, the

parametrizations used by Ross et al. (2006) and by Adduce

et al. (2012), which refer to the empirical entrainment relations

adopted in shallow water simulations of lock-release gravity

currents, supply values of Eof the same order of magnitude

of the present entrainment evaluations. In addition, in Jacobson

and Testik (2014) a further entrainment parametrization based

on both Fand Rnumbers is suggested. This parametrization,

applied to the numerical runs of this study, predicts values of E

ranging between 3 ×10−3−7×10−3and thus a better agree-

ment is found, if compared to Cenedese and Adduce (2010)

and Parker et al. (1987) entrainment parametrizations. Then, the

choice of a suitable entrainment parametrization in simpliﬁed

models simulating gravity currents, as shallow water models,

should be made with care depending on the unsteady or steady

nature of the ﬂow.

6 Variation of the dense current fractional area

As mentioned in the introduction, the investigation on mixing

during the slumping phase of lock-release gravity currents led

to contrasting results. According to Hallworth et al. (1996),

the gravity current starts to entrain ambient ﬂuid only dur-

ing the self-similar phase. However, other experimental studies

(Hacker et al. 1996; Jacobson & Testik 2013,2014; Nogueira

et al. 2013a,2014) found mixing during the slumping phase.

Fragoso et al. (2013) justiﬁed these diﬀerent ﬁndings in view

of the importance of the iso-density threshold chosen to deﬁne

the interface between the current and the ambient ﬂuid. In

fact, the ratio between the area of the current Aand the ini-

tial area of the lock ﬂuid A0decreases or increases versus

time, depending on the iso-density level chosen to deﬁne A.

The ratio A/A0versus x∗

fcan be used as a further indica-

tor of the entrainment in lock-release gravity currents follow-

ing the approach of Hacker et al. (1996) and Jacobson and

Testik (2014). A/A0calculated as in Hacker et al. (1996),

i.e. with ρ∗=5%, ρ∗=20%, ρ∗=40%, ρ∗=60%,

ρ∗=80% and ρ∗=100%, plus the reference value used

in this study ρ∗=2%, is shown in Fig. 11a–c. Ak/A0ver-

sus x∗

fare plotted varying rfor RUN2, RUN5 and RUN8 in

Fig. 11a–c, respectively. A vertical grey dashed line is plotted

in Fig. 11 to mark the end of RUN8. The same iso-density lev-

els are used to plot the spanwise-averaged dimensionless density

ﬁelds of Fig. 11d. In Fig. 11a–c an increase of the volume of

the current is detectable for the whole duration of all the experi-

ments with A1/A0, i.e. ρ∗=2%. This suggests the presence

of entrainment of ambient ﬂuid for the whole duration of the

experiments. The choice of values of the iso-density thres-

holds ρ∗>40% leads to opposite ﬁndings: the slope of the

curves is negative, then the volume of ﬂuid at higher density

decreases.

The extent of mixing within the dense current for the three

diﬀerent cases can be deduced, following the approach adopted

by Hacker et al. (1996), by the distribution of the diﬀerent curves

Ak/A0up to x∗

f6.5. In the r=2 case the mixing inside the

current is more homogeneous than in the r=0.5 case. In fact,

in Fig. 11a, the curves corresponding to A1,A2,A3,A4and

A5do not vary their slope, and all, except A5, increase as x∗

f

increases. In Fig. 11b and c A4decreases with x∗

fand it is

far from A1,A2and A3, indicating that the ambient ﬂuid is not

intruding into the core of the gravity current. In Fig. 11c both

A4/A0and A3/A0stop increasing with x∗

fat the end of the sim-

ulation, and thus the dense current entrains light ﬂuid only at

the leading edge of the current, and the interior part is essen-

tially unmixed. A similar trend was also found in the analysis

of the curves Ak/A0of Hacker et al. (1996). The results are

consistent with the plots of the density ﬁelds shown in Figs 4

and 5for r=2, g

0=0.15 m s−2and r=0.5, g

0=0.59 m s−2

cases.

The present numerical simulations agree with the results

obtained in the laboratory by Fragoso et al. (2013) and by

Hacker et al. (1996), showing that mixing occurs not only

during the self-similar, but also during the slumping phase.

However, in both previous investigations the experiments were

carried out only up to the end of the slumping phase, while

the present numerical results give a novel contribution to the

investigation of mixing occurring also during the self-similar

phase.

Journal of Hydraulic Research Vol. 54, No. 5 (2016) Entrainment and mixing in unsteady gravity currents 553

(a)

(b)

(c)

Figure 11 Ak/A0versus x∗

f: (a) RUN2, with r=2; (b) RUN5, with r=1; (c) RUN8, with r=0.5. Diﬀerent thresholds are used to deﬁne the edge of

the current: ρ∗=2% for A1,ρ∗=5% for A2,ρ∗=20% for A3,ρ∗=40% for A4,ρ∗=60% for A5,ρ∗=80% for A6and ρ∗=100%

for A7. (d) Spanwise averaged dimensionless density ﬁeld showing the same iso-density levels used to plot Ak/A0

7 Potential energies and irreversible mixing

Since the approaches shown in Section 5and 6are strongly

aﬀected by the density threshold used to deﬁne the interface

between the two layers, mixing within a lock-release grav-

ity current is herein studied by the energy budget method of

Winters et al. (1995). This method is not dependent on the

distinction between the volume of the dense current and the

volume of the ambient ﬂuid. For this reason, there is no need

to deﬁne any interface between the dense and the light ﬂuids

and this methodology was successfully applied in previous stud-

ies (Fragoso et al. 2013; Patterson, Caulﬁeld, McElwaine, and

Dalziel 2006). The energy budget method is based on the dis-

tinction between the adiabatic processes, which can alter the

initial potential energy Epof the ﬂow without exchanges of mass

or heat, and the diabatic ones. The potential energy, Ep,ofthe

ﬂow is deﬁned as:

Ep(t)=gV

ρ(x,y,t)ydV(23)

where ρis the mean density ﬁeld; xand yare respectively

the streamwise and the vertical coordinates and Vis the entire

volume of the ﬂuid, i.e. both ambient and dense ﬂuids. The

background potential energy Ebis deﬁned as:

Eb(t)=gV

˜ρ(x,y,t)ydV(24)

where ˜ρis the density ﬁeld adiabatically rearranged to the

conﬁguration of minimum of the potential energy, i.e. with the

ﬂuid particles redistributed in the domain in a perfectly stable

horizontally stratiﬁed setting. Thus Ebis resulted from a spa-

tial redistribution, attainable through adiabatic processes, of the

mean density ﬁeld from ρto ˜ρ, where the particles of ﬂuid

are set in the minimum state of the potential energy.

Adiabatic processes can modify the potential energy Ep

through a redistribution of ρwithout altering the background

energy (Winters et al. 1995). The diﬀerence between Epand Eb

is deﬁned as the available potential energy Ea:

Ea(t)=Ep−Eb(25)

Eais the “potential energy released in an adiabatic transition

from ρ(x,y,t)to ˜ρ(x,y,t),” without altering the probability

density function of density. “Furthermore, changes in the poten-

tial energy of the background state, Eb, are direct measure of the

energy expended in mixing the ﬂuid ” (Winters et al. 1995).

The energy budget is reported in Fig. 12 for RUN1. A simi-

lar behaviour was observed in the other cases analysed (not

shown). At the beginning of all the simulations, the variation

of potential energy Epis totally imputable to the conversion

to kinetic energy: Epand Eaare nearly coincident and Ebis

almost zero. Soon after, Ebincreases, implying the presence of

low but non-zero irreversible mixing. Thus, the ambient ﬂuid

554 L. Ottolenghi et al. Journal of Hydraulic Research Vol. 54, No. 5 (2016)

Figure 12 Energy budget for RUN1 (g

0=0.15 m s−2,r=2): poten-

tial energy (black solid line), background potential energy (grey solid

line) and available potential energy (dashed line) versus (x∗

f−x0)/H

trapping due to Kelvin–Helmholtz billows alters the energy bud-

get, slightly changing the background potential energy. During

the subsequent times Ebcontinues to increase as the front of

the current advances, while Eadecreases and separates from the

curve of Ep: conversion from the potential energy to the kinetic

energy occurs during the slumping phase. On the other hand,

the increase of Ebduring the constant velocity phase proves

the presence of irreversible mixing of ﬂuid elements. Kelvin–

Helmholtz billows cause a reversible stirring of the dense ﬂuid

into the body of the current and a permanent mixing with the

ambient ﬂuid. In Fig. 12 the curves of Eband Eacross with each

other and Ebbecomes greater than Ea: beyond this point the

variation of Epis caused more by the irreversible mixing than

by the transfer to kinetic energy.

To emphasize the rate of mixing processes, the evolution of

the time derivatives dEb/dtis analysed. In Fig. 13 the quantity

(dEb/dt)∗is made dimensionless with Eb(t=0) as a scale for

the energy and the total duration of each simulation as a time

scale. Cases with g

0=0.15, 0.29 and 0.59 m s−2are reported

in Fig. 13a–c, respectively. For increasing g

0, higher values of

(dEb/dt)∗are visible, indicating the increment of irreversible

mixing. Curves labelled with r=0.5 are always above those

with r=1 and r=2. This means that mixing processes are

more eﬃcient with the decrease of r. At the beginning of each

simulation, the curves rise steeply. After that, up to the end of the

slumping phase, (dEb/dt)∗ﬂuctuates around an almost constant

value. Finally, the decrease of d(Eb/dt)∗is observable during

the self-similar phase. This behaviour indicates the occurrence

of distinctive processes driving mixing during the diﬀerent ﬂow

regimes. In fact, as mentioned before, the generation of Kelvin–

Helmholtz instabilities characterize the ﬁrst phase, while the

existing Kelvin–Helmholtz billows start to lose their coherence

and collapse during the self-similar phase, when a maximum of

(dEb/dt)∗occurs. After the collapse of Kelvin–Helmholtz bil-

lows, the turbulent structures become more three-dimensional

and a decrease of (dEb/dt)∗is observed (Fig. 13).

8 Conclusions

In the present paper the dynamics of unsteady gravity cur-

rents were investigated by high resolution wall-resolving LES.

Laboratory experiments were also performed. The compar-

ison between numerical and experimental results showed a

reasonable agreement also in consideration of the inevitable

diﬀerences in the initial and boundary conditions. Nine diﬀer-

ent cases of lock-exchange gravity currents developing over

a horizontal surface were examined through both laboratory

experiments and numerical simulations. The study was aimed

at the analysis of entrainment and mixing under unsteady con-

ditions, achieved through the lock-release technique. The ﬂow

conditions were obtained, varying both the initial excess den-

sity and the aspect ratio rof the initial lock ﬂuid domain.

The range of parameters investigated allowed reproduction of

both the slumping and the self-similar regimes in the gravity

currents with vertically-oriented and square-shaped initial lock

ﬂuid domains, whereas the slumping phase only developed in

the cases with a horizontally-oriented initial lock ﬂuid domain.

The analysis of the time variation of the front position xf(t)

showed an increase in velocity of the front with increasing g

0

or decreasing r. In agreement with the literature, a universal

behaviour of the dimensionless x∗

f(T∗) was observed for all

cases examined. The aspect-ratio of the lock ﬂuid was shown

to aﬀect the time development of the gravity current and the

associated mixing. In the r=0.5 and r=1 runs, the presence

of an unmixed region in the core of the gravity current was

observed, and the analysis of the dense current fractional area

in dependence on the density threshold conﬁrmed this trend.

The three-dimensional behaviour of the gravity currents was

investigated by analysing the evolution in time of the density

(a) (b) (c)

Figure 13 dEb/dtversus (xf−x0)/Hfor all the simulations. (a) g

0=0.15 m s−2cases; (b) g

0=0.29 m s−2cases; (c) g

0=0.59 m s−2cases.

Runs with r=2 (solid grey lines), cases with r=1 (dashed lines) and r=0.5 cases (solid black lines)

Journal of Hydraulic Research Vol. 54, No. 5 (2016) Entrainment and mixing in unsteady gravity currents 555

iso-surface ρ∗=2%. Kelvin–Helmholtz billows and lobe-

and-cleft instabilities were clearly detected. When the Kelvin–

Helmholtz billows collapsed, three-dimensional disturbances

were generated. They propagated up-stream until they occupied

the entire interface between the current and the ambient ﬂuid.

It was found that in the gravity currents with a horizontally-

oriented initial lock ﬂuid domain (small r) the three-dimensional

structures at the interface developed earlier and were stronger.

This was attributed to the higher values of the bulk velocity of

the current for the small rcases, which produced a faster and

more energetic development of the interfacial mixing layer.

These features had an impact on the entrainment and mix-

ing properties of the gravity currents. In particular, mixing was

evaluated using three diﬀerent approaches. The ﬁrst two meth-

ods are based on the evaluation of two entrainment parameters

(Ebulk and Einst) and on the variation in time of the fractional

area of the dense current; the third approach considers the evo-

lution of the background potential energy. In agreement with

previous studies, it was found that the entrainment parameters

and the variations of the fractional area of the gravity current

are sensitive to the selection of the iso-density threshold used

to deﬁne the interface between the dense and the ambient ﬂu-

ids. This limit was overcome by the third method, which is

based on energetic considerations. The application of the lat-

ter clearly showed the presence of irreversible mixing during

both the slumping and the self-similar phases. In the entrain-

ment analysis, peaks in Einst were more pronounced in the r=2

cases with respect to the r=0.5 cases, but the total amount of

ambient ﬂuid entrained by the current resulted higher in the

r=0.5 cases. The analysis of the density ﬁelds revealed the

determinant role of Kelvin–Helmholtz instabilities in mixing

dynamics. In the r=2 cases, the billows trapped ambient ﬂuid

into the body of the current and generated irreversible mixing

during their collapse. When billows increased in dimension and

in intensity, patches of ambient ﬂuid were caught up, the gravity

current increased in volume, and peaks of Einst were observed. In

cases with r≤1 the Kelvin–Helmholtz billows were conﬁned

inside the body of the current and were more stable. Conse-

quently, peaks of Einst were less evident, but the population of

instabilities was able to produce globally a greater amount of

mixing. The value of the global volume entrained by the gra-

vity current, E, was also dependent on r. Both the numerical and

the experimental data indicated that the r=0.5 cases entrained

more ambient ﬂuid than the r=1 and the r=2 cases. Finally,

the importance of a suitable entrainment parametrization was

highlighted for those simpliﬁed models, such as shallow water

models, which are not able to solve the small scale mixing

process.

The analysis of the temporal derivatives of Eb(which are use-

ful to detect the occurrence of irreversible mixing processes)

highlighted the eﬀects of rand g

0: mixing was enhanced by

increasing g

0and gravity currents with r=0.5 showed greater

values of (dEb/dt)∗than the gravity currents with r=2. This is

in agreement with the discussion of the entrainment parameters.

In the r=2 cases, an increase of (dEb/dt)∗during the transition

to the self-similar phase was observed, revealing the presence

of more eﬃcient processes of permanent mixing related to the

collapse of Kelvin–Helmholtz billows.

Funding

C.A. wishes to thank Cineca, which supported in part this

research by the ISCRA 2013 program through the research

project “LES investigation of 3D and up-sloping density cur-

rents” and the ISCRA 2014 program through the research

project “LES investigation of 3D lock-release gravity currents”.

Notation

A0,Ai=area of the current at t0and at a general

time tiin the vertical x–yplane (m2)

Ak=area of the current deﬁned by a ﬁxed iso-

density threshold

d=spanwise dimension of the domain (m)

E=bulk entrainment parameter evaluated at

the end of the simulation (–)

Ea=available potential energy (kg m2s−2)

Eb=background potential energy (kg m2s−2)

Ebulk =bulk entrainment parameter (–)

Einst =instantaneous entrainment parameter (–)

Ep=potential energy (kg m2s−2)

F=Ubulk/g

mh0/2, bulk Froude number (–)

Fslump =Ubulk(t=ts)/g

0h0, Froude number at

the end of the slumping phase (–)

g=gravitational acceleration (m s−2)

g

0=g(ρ1−ρ0)/ρ0, initial reduced gravity

(m s−2)

g

end =ﬁnal reduced gravity (m s−2)

g

m=(g

0+g

end)/2, mean reduced gravity

(m s−2)

H=depth of the ambient ﬂuid (m)

h0=initial depth of the dense current (m)

k=molecular diﬀusivity for salt water

(m2s−1)

ks=molecular salt diﬀusivity (m2s−1)

L=length of the experimental tank (m)

Lx,Ly,Lz=length of the numerical domain in the x,y,

z-directions (m)

p=hydrodynamic pressure ( kg m−1s−2)

Qei =Vi/ti, bulk entrainment discharge

(m3s−1)

Qei_inst =instantaneous entrainment discharge

(m3s−1)

R=Ubulk(h0/2)/ν, Reynolds number (–)

Rb=ubH/ν, buoyancy Reynolds number (–)

r=h0/x0, aspect ratio of the lock volume (–)

Sc =k/ν, Schmidt number (–)

556 L. Ottolenghi et al. Journal of Hydraulic Research Vol. 54, No. 5 (2016)

Si=xﬁd, interface between the dense and the

ambient ﬂuids in the x–zplane (m2)

s=salinity (psu)

s0=reference value of salinity (psu)

T∗=tub/x0dimensionless time (–)

t=time variable (s)

t0,ti=initial time and general time (s)

ts=slumping time (s)

Ubulk =bulk velocity of the dense current (m s−1)

Ubsl =xs/ts, bulk velocity at the end of the

slumping phase (m s−1)

ub=g

0H, buoyancy velocity (m s−1)

ui=velocity component in the i-direction

V0,Vi=volume of the current at t0and at a general

time ti(m3)

Wei =Qei/Si, bulk entrainment velocity (m s−1)

Wei_inst =instantaneous entrainment velocity (m s−1)

x0=distance of the removable gate from the

left wall of the tank (m)

xf=front location of the gravity current (m)

xfex =experimental front location (m)

xfnum =numerical front location (m)

x∗

f=(xf−x0)/x0, dimensionless front loca-

tion of the gravity current (–)

xi=space variable in the i-direction (m)

xs=distance travelled by the current in the

slumping phase (m)

x,y,z=space variable in the streamwise, vertical

and spanwise directions (m)

ti=general time step (s)

x+,y+,z+=dimensionless spacing variables (–)

Vi=increase in volume at a general time ti

(m3)

β=salinity cubic contraction coeﬃcient

(psu−1)

=(ρ1−ρ0)/ρ0, initial excess density (–)

lj=SGS salinity ﬂux (m psu s−1)

ν=kinematic viscosity of the water (m2s−1)

ρ=density (kg m−3)

ρ=Boussinesq density variation (kg m−3)

ρ∗=(ρ −ρ0)/(ρ1−ρ0), dimensionless den-

sity ﬁeld (–)

ρ0=initial density of the ambient ﬂuid

(kg m−3)

ρ1=initial density of the dense ﬂuid (kg m−3)

˜ρ=density ﬁeld adiabatically rearranged to

the minimum of Ep(kg m−3)

τij =SGS momentum ﬂux (m2s−2)

ORCID

Claudia Adduce http://orcid.org/0000-0002-0734-9569

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