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J. Fluid Mech. (2018), vol. 840, pp. 579–612. c Cambridge University Press 2018 doi:10.1017/jfm.2017.917 579 Critical regime of gravity currents flowing in non-rectangular channels with density stratification L. Chiapponi 1, , M. Ungarish 2 , S. Longo 1 , V. Di Federico 3 and F. Addona 1 1 Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy 2 Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel 3 Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Università di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy (Received 14 February 2017; revised 24 October 2017; accepted 18 December 2017) We present theoretical and experimental analyses of the critical condition where the inertial–buoyancy or viscous–buoyancy regime is preserved in a uniform-density gravity current (which propagates over a horizontal plane) of time-variable volume V = qt δ in a power-law cross-section (with width described by f (z) = bz α , where z is the vertical coordinate, b and q are positive real numbers, and α and δ are non-negative real numbers) occupied by homogeneous or linearly stratified ambient fluid. The magnitude of the ambient stratification is represented by the parameter S, with S = 0 and S = 1 describing the homogeneous and maximum stratification cases respectively. Earlier theoretical and experimental results valid for a rectangular cross-section (α = 0) and uniform ambient fluid are generalized here to a power-law cross-section and stratified ambient. Novel time scalings, obtained for inertial and viscous regimes, allow a derivation of the critical flow parameter δ c and the corresponding propagation rate as Kt β c as a function of the problem parameters. Estimates of the transition length between the inertial and viscous regimes are also derived. A series of experiments conducted in a semicircular cross-section (α = 1/2) validate the critical values δ c = 2 and δ c = 9/4 for the two cases S = 0 and 1. The ratio between the inertial and viscous forces is determined by an effective Reynolds number proportional to q at some power. The threshold value of this number, which enables a determination of the regime of the current (inertial–buoyancy or viscous–buoyancy) in critical conditions, is determined experimentally for both S = 0 and S = 1. We conclude that a very significant generalization of the insights and results from two-dimensional (rectangular cross-section channel) gravity currents to power-law cross-sections is available. Key words: geophysical and geological flows, gravity currents, stratified flows † Email address for correspondence: [email protected] https://www.cambridge.org/core/terms . https://doi.org/10.1017/jfm.2017.917 Downloaded from https://www.cambridge.org/core . Universita' di Parma. Dipartimento di Filologia Classica e Medievale , on 15 Feb 2018 at 12:40:59, subject to the Cambridge Core terms of use, available at
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J. Fluid Mech. (2018), vol. 840, pp. 579–612. c© Cambridge University Press 2018doi:10.1017/jfm.2017.917

579

Critical regime of gravity currents flowing innon-rectangular channels with

density stratification

L. Chiapponi1,†, M. Ungarish2, S. Longo1, V. Di Federico3 and F. Addona1

1Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze,181/A, 43124 Parma, Italy

2Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel3Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Università di

Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy

(Received 14 February 2017; revised 24 October 2017; accepted 18 December 2017)

We present theoretical and experimental analyses of the critical condition wherethe inertial–buoyancy or viscous–buoyancy regime is preserved in a uniform-densitygravity current (which propagates over a horizontal plane) of time-variable volumeV = qtδ in a power-law cross-section (with width described by f (z) = bzα, wherez is the vertical coordinate, b and q are positive real numbers, and α and δ arenon-negative real numbers) occupied by homogeneous or linearly stratified ambientfluid. The magnitude of the ambient stratification is represented by the parameterS, with S = 0 and S = 1 describing the homogeneous and maximum stratificationcases respectively. Earlier theoretical and experimental results valid for a rectangularcross-section (α = 0) and uniform ambient fluid are generalized here to a power-lawcross-section and stratified ambient. Novel time scalings, obtained for inertialand viscous regimes, allow a derivation of the critical flow parameter δc and thecorresponding propagation rate as Ktβc as a function of the problem parameters.Estimates of the transition length between the inertial and viscous regimes are alsoderived. A series of experiments conducted in a semicircular cross-section (α = 1/2)validate the critical values δc= 2 and δc= 9/4 for the two cases S= 0 and 1. The ratiobetween the inertial and viscous forces is determined by an effective Reynolds numberproportional to q at some power. The threshold value of this number, which enables adetermination of the regime of the current (inertial–buoyancy or viscous–buoyancy) incritical conditions, is determined experimentally for both S=0 and S=1. We concludethat a very significant generalization of the insights and results from two-dimensional(rectangular cross-section channel) gravity currents to power-law cross-sections isavailable.

Key words: geophysical and geological flows, gravity currents, stratified flows

† Email address for correspondence: [email protected]

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580 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

1. IntroductionGravity (or density) currents (GCs) spread out every time that a fluid propagates

into an ambient and the flow is driven by a density difference. This type of flow hasboth geophysical and industrial applications, such as saltwater wedges in estuaries, oilspills in the ocean, magma and lava flows, dredge disposal and industrial gas release;numerous examples are illustrated by Simpson (1997).

A stringent classification of gravity currents is related to the main balancedominating the flow, either inertial–buoyancy or viscous–buoyancy.

In a two-dimensional geometry, both inertial and viscous currents of volume qtδ,q > 0, δ > 0, display self-similar propagation of the form xN = Kitβi , where i = I, Vfor the inertial and the viscous regimes respectively. In general, βi increases with δ(Didden & Maxworthy (1982), Huppert (1982); see also Ungarish (2009) § 4.2). Sincethe GC is a strongly time-dependent phenomenon, a downstream change of dynamicregime may occur at some distance from the injection section. For the more commonoccurrence δ < 7/4 (subcritical case), the transition is between the inertial and viscousregimes, and is manifested by a decrease of the β exponent. There is, however, aninteresting ‘critical’ exception: the theory predicts that for δ= 7/4, βI =βV = 5/4, andhence a regime transition is impossible. A two-dimensional (2-D) current sustained bythe critical exponent of the influx rate δc = 7/4 is expected to maintain the originalflow regime for a very long time; in practice, the nature of the flow will eventually bechanged by some obstacle, or by the turn-off of the injection source. This fixed-forceregime is characterized by a fixed effective Reynolds number Ree= (q2/g′)2/3/ν, whereg′ is the reduced gravity and ν is the kinematic viscosity (we note in passing that thethird power of the inverse of this number has been defined as the Julian number, J,but we think that Ree is a more insightful parameter). These theoretical predictionswere confirmed by stringent experimental tests conducted by Maxworthy (1983).

The ‘critical’ fixed-force regime behaviour has been considered as a strongachievement of gravity current investigations. Here, the theoretical tools for inertialand viscous currents combine with experimental skill into sharp unexpected insightsand results. The available theoretical formulation for critical gravity currents isrestricted to channels of rectangular, or infinitely wide, cross-section. The objectiveof the present investigation is to determine whether and how the ‘critical’ effect ofthe constant force regime can be generalized to gravity currents in non-rectangularchannels.

An extension to channels of non-rectangular cross-section represents an importanttheoretical advancement as it implies an increase in domain dimensionality andincorporates three-dimensional (3-D) effects (Longo et al. 2015b, 2016b). Furthermore,the inclusion of stratification effects in the ambient fluid extends the investigation toan important class of non-homogeneous fluids, whose association with gravity flowsproduces striking phenomena (Longo et al. 2016a).

For inertial currents, the effects of non-rectangular cross-section channels wereinvestigated by Monaghan et al. (2009), Marino & Thomas (2011) and Zemach& Ungarish (2013); the effect of a linear stratification was incorporated into themodel by Ungarish (2015). Viscous gravity currents propagating into non-rectangularcross-sections filled with homogeneous ambient fluid were modelled by Takagi &Huppert (2007, 2008); to the best of our knowledge, the joint effect of viscosityand stratification has not been considered in the literature. A modelling effortto incorporate non-Newtonian effects was undertaken by Longo, Di Federico &Chiapponi (2015a). See Ungarish (2018) for a short review.

The simplified thin-layer equations (of shallow-water type for an inertial–buoyancycurrent and lubrication type for a viscous–buoyancy current) are partial differential

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Gravity currents in critical condition 581

with respect to time t and length x, and in general require numerical solution.However, it turns out that for channels of power-law cross-section f (z) = bzα, theseequations admit similarity solutions, and the resulting propagation is of the formxN = Ktβ for a variety of boundary conditions. (Here, z is the vertical coordinateand K is a coefficient depending on several parameters of the current; the 2-Drectangular cross-section is just a particular case, corresponding to α = 0.) Thistype of solution is a very convenient (and to the best of our knowledge the onlyrigorous) tool for analytical investigation of long-time (or long-distance) propagationof gravity currents. We shall therefore base the present investigation on this typeof flow. An additional advantage of the similarity solution is that it is compatiblewith the versatile box-model approximation. The ‘box’ shape is actually a similaritypostulate, and in all tested cases we found that this rough approximation reproducesthe correct tβ propagation.

There is evidence in general (both theoretical and experimental) that self-similarflow is an attractor of the gravity current long-time propagation in both inertialand viscous regimes. See Takagi & Huppert (2008), Longo et al. (2013, 2015a),Zemach & Ungarish (2013) and Ungarish (2015), where self-similarity is verified fora variety of conditions (2-D flows in rectangular and non-rectangular cross-sections,axisymmetric flows, viscous Newtonian and non-Newtonian fluid flows). In view ofthese considerations, the self-similar solutions are the major analytical tools used inour study. We note that the 2-D critical-flow predictions that are generalized herewere obtained by box-model approximations.

The present work develops theoretically and tests experimentally the ‘critical’current generalized to non-rectangular channels of power-law shape and to a linearlystratified ambient fluid with maximum stratification. With this overarching objective,several novel self-similar scalings valid for the inertial and viscous regimes are derivedin passing; these results generalize earlier derivations valid in less general conditions(2-D flow and homogeneous fluid). Moreover, the effective Reynolds number and itsthreshold value controlling the balance in critical conditions are discussed in detail.The structure of the paper is as follows.

Section 2 examines the inertial regime and presents novel scalings for the lengthand thickness of the gravity current, extending earlier results valid for constant volume(Zemach & Ungarish 2013; Ungarish 2015) to variable volume. Section 3 considersthe viscous regime and presents scalings valid for ‘wide’ and ‘narrow’ cross-sections.Section 4 analyses the transition length between the inertial and viscous regimes, anddiscusses its behaviour as a function of the strength of the injection, the channelshape and the stratification. Section 5 discusses the critical conditions, based on thederivations of §§ 2 and 3. First, the critical inflow parameter δc and the correspondingpropagation rate βc are evaluated as a function of the problem parameters; then, thescalings of the front position with the multiplicative coefficient q in the expression ofthe volume of the current are derived as a means to detect variations in the regimeof the current during experimental runs. The transition Reynolds number in a circularcross-section is derived. Section 6 is devoted to experimental work performed tovalidate theoretical findings. A series of experiments were performed with critical δin a circular channel using both homogeneous and linearly stratified ambient fluid,in order to assess the theoretical predictions and to determine the value of Ree thatseparates the regimes of viscous–buoyancy and inertial–buoyancy. The experimentalapparatus, the tests conducted and their interpretation are described sequentially.Section 7 reports the summary and the main conclusions and perspectives for futurework. The numerical integration and the detailed computations of the scaling of thefront position are included in the appendices.

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582 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

x

z

H

0

Ambient

(a)

(b) (c)

g

Q

y

C

z

g

H

h

A

r

y

z

FIGURE 1. (Colour online) A sketch of the current advancing in a homogeneous (S= 0)or linearly stratified ambient fluid with ρb = ρc (S= 1): (a) side view; (b) cross-view; (c)two power-law cross-sections f (z)∝ zα for α = 1/2 and α = 2.

2. Similarity solutions for the inertial regimeWe consider a gravity current propagating in a channel of constant cross-section,

whose geometry is represented by the width function described by f (z). The channelis occupied by an ambient fluid, either homogeneous or non-homogeneous; seefigure 1. We introduce the thickness of the current h(x, t) (the position of theinterface between intruding and ambient fluid) and its average velocity u(x, t) overits area A. In many cases of interest, the intruding current, of characteristic thicknesshN , is thin with respect to the thickness H of the ambient fluid, and the densitydifferences between the current and the ambient (of densities ρc and ρa respectively)are relatively small. We also assume that the typical width of the current is notsignificantly smaller than the thickness, i.e. f (hN) & hN . This leads to the one-layershallow-water (SW) Boussinesq model, incorporating two main simplifications inthe high-Reynolds-number equations of motion. First, the ambient fluid is taken tobe at rest (ua = 0), neglecting the ‘return’ flow in the ambient above the current.This is tantamount to assuming h/H → 0. However, for the calculation of Fr (theFroude-number jump condition) at the nose, a finite ratio hN/H is adopted. Thesecond assumption is that the density difference ρc − ρa is neglected in the inertiaterms, but taken into account into the term driving the motion, which includes thereduced gravity g′.

We incorporate in the theory an additional physical effect, the density stratificationof the ambient fluid. Following Ungarish (2015), we consider the case of a linearlystratified ambient, whose density increases from ρ0 at the top, z = H, to ρb at thebottom, z= 0,

ρc = ρ0(1+ ε ), ρa(z)= ρ0

[1+ ε S

(1−

zH

)], (2.1a,b)

whereε =

ρc − ρ0

ρ0, S=

ρb − ρ0

ρc − ρ0, g′ = ε g. (2.2a−c)

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Gravity currents in critical condition 583

The additional parameter S ∈ [0, 1] represents the magnitude of the stratificationin the ambient fluid. Two limit cases are possible: when S = 0, the ambient fluidis homogeneous; when S = 1, the maximum stratification takes place and ρc = ρb.(The stratified ambient sustains internal waves with frequency N = (Sg′/H)1/2 andmaximum speed NH/π in a rectangular section, and N r/(2

√2) in a circular section

of radius r, which are outside the scope of this paper.)According to the thin-layer assumption, the pressure distribution in both the ambient

fluid and the intruding current is hydrostatic, i.e. ∂pi/∂z=−ρig, where pi, i= a, c, arethe relevant pressure fields. Taking into account the continuity of the pressure at theinterface z= h, one obtains

pa(z)=−ρ0g[1+ ε S

(1−

z2H

)]z+C, (2.3)

pc(z)=−ρ0g(1+ ε )(z− h)+ pa(h), (2.4)

where C is a constant. Equation (2.4) yields the x-component of the pressure gradientas

∂pc

∂x= g′ρ0

(1− S+ S

hH

)∂h∂x, (2.5)

which is constant over the cross-section A of the intruding current. We now averagethe x-momentum inviscid equation over A, taking into account inertial and buoyancyterms. The former, ρc Du/Dt, is unchanged with respect to the homogeneous case; thelatter is given by (2.5). Within the framework of the Boussinesq approximation (ρc≈

ρ0), the momentum equation is then obtained as

∂u∂t+ u

∂u∂x=−g′

(1− S+ S

hH

)∂h∂x. (2.6)

The mass balance equation (Ungarish 2015) reads as

∂h∂t+ u

∂h∂x+

Af (h)

∂u∂x= 0. (2.7)

Hereafter, we use a tilde to indicate a dimensionless variable. We scale horizontallengths along the channel with x0, vertical lengths with h0, width with f (h0), speedwith U and time with T , where

U =(g′h0

)1/2, T = x0/U. (2.8a,b)

The scales x0, h0 are problem-dependent. For the fixed-volume current, the length andheight of the lock are appropriate. For the influx current, it is convenient to definex0= h0= L, where L is the height of the ambient H, while for the constant-flux case,it may also be convenient to take for L the height of the source. For the experimentsdiscussed in this paper, L=H.

Motivated by the analysis of the characteristics of the hyperbolic system (2.6)–(2.7),we specify the boundary condition at the nose, x = xN , as the discontinuous jumpcondition

uN = FrUh1/2N Ψ 1/2, (2.9)

where FrU is the ‘Froude number’, given by (Ungarish 2012)

Fr2U =

2(1− ϕ)1+ ϕ

[1− ϕ +

1hAT

∫ h

0zf (z)dz

], ϕ = A/AT, (2.10a,b)

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584 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

AT is the total area of the channel occupied by the fluid, and in which the stratificationcoefficient

Ψ =

∫ hN

0[(pc − pa)S] f (z) dz∫ hN

0[(pc − pa)S=0] f (z) dz

(2.11)

is the ratio of pressure force over the nose with and without stratification. Using (2.3)–(2.4), we obtain

Ψ = 1− S[

1−12

hN

H(1+ γ )

], (2.12)

where the coefficient γ is given by

γ =

∫ hN

0

zhN(hN − z)f (z) dz∫ hN

0(hN − z) f (z) dz

. (2.13)

The actual value of γ depends on the cross-section shape: for the 2-D case, γ = 1/3,while for the circular cross-section, γ ≈ 0.43. As regards the value of Ψ , it is notedthat the stratification slows down the propagation: for S= 0, Ψ = 1, then Ψ decreaseswith increasing S.

The theoretical formulation is completed by the integral mass balance equation,expressing the injected volume V as

V =∫ xN

0A(h) dx= q tδ, (2.14)

where q> 0, δ> 0; when the injected volume V is fixed and equal to V0, then δ= 0and q= V0.

For mathematical convenience, we apply the transformation ξ = x/xN(t), whichmaps the x ∈ [0, xN(t)] domain of flow into the constant ξ ∈ [0, 1] (for more details,see Ungarish (2009) § 2.3). For the A(ξ , t), u(ξ , t) variables, the previous equationstransform according to(

∂t

)x

=

(∂

∂t

− ξxN

xN

(∂

∂ξ

)t

,

(∂

∂x

)t

=1xN

(∂

∂ξ

)t

, (2.15a,b)

where the subscripts denote the ‘fixed’ variable in partial differentiation and the upperdot means time derivative.

After significant propagation time, the influence of the initial conditions diminishes,and the ratio hN/H is sufficiently small to justify a constant Fr=Fr(0) approximation.We then seek a similarity solution of the form (see Zemach & Ungarish 2013;Ungarish 2015)

xN =K t βI , h(ξ , t)=Ω(t)H(ξ), u(ξ , t)= ˜xNU(ξ), (2.16a−c)

where the upper dot means time derivative, K, βI are positive constants and Ω(t),H(ξ), u(ξ , t) are unknown functions. The boundary condition on the velocity isU(1)= 1 and the integral mass conservation yields

V =∫ xN

0A(h) dx= xN

∫ 1

0A[Ω(t)H(ξ)] dξ = q t δ, (2.17)

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Gravity currents in critical condition 585

where δ> 0. The analysis proves that such a similarity solution exists only under thefollowing restrictions (the exception is the constant-flux δ= 1 current). First, we mustuse a power-law cross-section, f (z)= bzα, where b has dimension [L1−α

] (the standardf (z)= 1 included, with V0= 1/(α+ 1)). With α= 1/2 and b= 2

√2r, this formulation

captures approximately also a semicircular cross-section of radius r. For a power-lawcross-section, A(h) = b h 1+α/(1 + α), and the integral in (2.17) attains a self-similarform. It is convenient to consider a rectangle of width b as a power-law profile withα= 0 (the formal jump of f (z) to 0 at z= 0 is insignificant in our analysis). The planecase can be analysed as a rectangular case with the assumption that b is much largerthan the current depth. Second, the ambient is either homogeneous, S = 0, or has alinear density stratification under the special case of maximum stratification, S = 1.This is needed to obtain the similarity form of the uN ≡ ˜xN condition. Under theseconditions, manipulation of the momentum balance equation, the volume conservationequation and the boundary conditions at the nose yields the following results.

(i) For S= 0,

Ω(t)= (˜xN)2, H(1)=

1Fr2

, βIh =2α + 2+ δ

2α + 3, (2.18a−c)

and the continuity and momentum equations are

(U − ξ)H′

H+

1α + 1

U ′ = 2(

1βIh− 1), H′ + (U − ξ)U ′ −

(1βIh− 1)U = 0.

(2.19a,b)

(ii) For S= 1,

Ω(t)= ˜xN, H(1)=(

2H1+ γ

)1/21Fr, βIs =

α + 1+ δα + 2

, (2.20a−c)

and the continuity and momentum equations are

(U − ξ)H′

H+

1α + 1

U ′ =1βIs− 1,

1

2H(H2)′ + (U − ξ)U ′ −

(1βIs− 1)U = 0.

(2.21a,b)Here, βIh and βIs stand for β in the inertial–buoyancy regime for currents in thehomogeneous (h) and stratified (s) ambient fluid respectively. These time exponentsare listed later in table 1, for comparison with values obtained in the viscous regime.Letting p= (α + 1)(2− S), (2.17) gives

K =

[(α + 1)q

pI

∫ 1

0Hα+1(ξ) dξ

)−1]1/(1+p)

. (2.22)

Simple analytical solutions are obtained for two important cases, δ = 0, 1. First, forthe fixed-volume current δ= 0, we notice that the solution U = ξ fulfils the continuityequation, while the integration of the momentum equation produces H(ξ). The valueof K is then determined from (2.22). For more details, see Zemach & Ungarish (2013)§ V and Ungarish (2015) § 5 (note the rescaling (5.1) there). Second, for the constant-influx current δ = 1, we find βI = 1, and then notice that the slug-like propagation

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586 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

Ambient fluid Viscous ‘wide’ Viscous ‘narrow’ Inertial,section, 0 6 α 6 1 section, α > 1 α > 0

Homogeneousα + 1+ 3δ

5+ 2αα(δ + 1)+ δ(α + 1)+ 1

4α + 32α + 2+ δ

2α + 3

Linearly stratified, S= 1α + 1+ 4δ

6+ 2α2δ + 1

4α + 1+ δα + 2

TABLE 1. Asymptotic self-similar solutions: values of the time scaling exponent β of thefront position, xN ∝ t β , for ‘narrow’ and ‘wide’ cross-sections respectively, advancing ina homogeneous (S= 0) and a linearly stratified ambient fluid with maximum stratification(S= 1).

U = 1 and H=H(1)= 1 satisfies the continuity and momentum equation. For othervalues of δ, numerical solutions for H, U subject to the conditions at ξ = 1 can beattempted.

We note that the self-similar solution for the standard 2-D (rectangular orunbounded) channel is recovered for α = 0. Here, we present a significant extension.The novelty is that self-similar currents can occur only in power-law cross-section, inparticular for a rectangle, α = 0, semicircle, α = 1/2 (approximately), or

∨triangle

α = 1. With hindsight, this is not surprising, because the self-similar distribution ofthe volume needs a compatible behaviour of the cross-section area.

In particular, for fixed-volume currents in homogeneous ambient, the rate ofpropagation β increases with α from the classical 2/3 towards 1. In these cases, thetransition from the slumping to the self-similar phase is less pronounced, and thismay strengthen the impression of propagation with almost constant speed over a longdistance.

A notable exception is the constant-flux δ = 1 current. In this case, Ω = const.,βI = 1, and some of the previous restrictions can be relaxed. We find that the slug-likeself-similar propagation with constant hN, uN is an exact solution for a general f (z),for both Boussinesq and non-Boussinesq systems. Moreover, this solution is also validfor the two-layer model because Fr is a constant due to the constant hN ; see Longoet al. (2015b).

The main deficiency of the similarity solutions is the vague connection withrealistic initial/boundary conditions of the current. In this respect, there is nodifference between the standard and the more general f (z) cases. In particular, inthe lock-released current, the shift of t by a constant to a ‘virtual origin’ does notaffect the solution.

We note that strong influx, δ > 1, may cause difficulties. For δ > 1, we obtain βI > 1for both the S=0 and 1 cases, i.e. the current accelerates. The coefficient ω=1/βI−1,which appears in the (reduced) continuity and momentum equations, becomes negative.We thus expect a change of behaviour between δ < 1 and δ > 1 cases. This is moreevident after a manipulation of the equations as follows.

The governing equations can be rewritten as follows.(i) For S= 0,

H′Q=ω [U − 2(α + 1)(U − ξ)] , U ′Q=ω(α + 1)[

2− (U − ξ)UH

], (2.23a,b)

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Gravity currents in critical condition 587

where

Q=Q(ξ)= 1− (α + 1)(U − ξ)2

H, ω=

1βIh− 1, (2.24a,b)

with the conditions U = 1,H= 1/Fr2,U ′= 2ω(α+ 1),H′=ω at ξ = 1. The integrationis performed backwards to ξ = 0. When influx is present, U(0),H(0)> 0; this impliesthat Q must decrease from 1 to smaller values. If (1 + α)(U − ξ)2/H = 1 at someinternal point, a discontinuity of H, U appears. The implication is that the boundaryconditions from the nose, ξ = 1, cannot influence the current for smaller values of ξ ;the influx is so strong that the conditions from the source prevail in the tail.

(ii) For S= 1, we introduce

κ =H/(2H)1/2, Q=Q(ξ)= 1− (α + 1)(U − ξ)2

2κ2, ω=

1βIs− 1. (2.25a−c)

The governing equations read as

κ ′Q=ω

2κ[U − (α + 1)(U − ξ)] , U ′Q= (α + 1)ω

[1−

U(U − ξ)2κ2

], (2.26a,b)

with U =1, κ=[(1+γ )1/2Fr]−1, κ ′=ω/2,U ′=ω(α+1) at ξ =1. Again, we encounterthe possibility of a singularity when Q= 0 at some ξ < 1.

A numerical integration has been performed in order to demonstrate that the self-similar solution is an attractor; see appendix A for some results for the two criticalcases with S= 0 and S= 1.

3. Similarity solutions for the viscous regimeThe similarity solutions discussed in the following for the viscous regime are the

counterparts of those discussed in the previous section for the inertial regime and areneeded to determine the critical condition. The present section recapitulates earlierresults for constant-volume viscous gravity currents in a power-law f (z)= bzα cross-section filled with homogeneous ambient fluid, and derives novel scalings for moregeneral cases including variable-volume currents and a linearly stratified ambient fluid.Typically, the current initiates the motion in an inertial (inviscid) regime, with a largeReynolds number Re; using the lock-release x0,h0 horizontal and vertical length scales,we define Re=Uh0/ν, where U= (g′h0)

1/2. After some significant distance xV (scaledwith x0, the subscript V stands for ‘viscous’), the long and thin current becomesviscous-dominated.

3.1. Current of constant volume advancing into a homogeneous ambient (S= 0)Viscous flow in a power-law cross-section was considered by Takagi & Huppert (2007)for the special case of constant volume and homogeneous ambient fluid. The no-slipconditions on the bottom and sidewalls activate the shear ν∇2u, which balances thebuoyancy driving force (per unit volume) −g′∂h/∂x. Therefore, h has a negativeslope, and h= 0 at x= xN . The z= h boundary of the current is a shear-free surface,∂u/∂z = 0, which represents approximately a thick layer of ambient fluid (typically,viscous effects become relevant when the intruding current is very thin).

In the following, we scale x with x0, y and z with h0, speed with U= (g′h0)1/2 and

time with x0/U; b is scaled with h1−α0 and the volume with x0h2

0.

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588 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

In dimensionless form, the governing balances for momentum and continuity are

∂2u∂ y 2+∂2u∂ z 2=

(Re

h0

x0

)∂ h∂ x, (3.1)

f (h)∂ h∂ t+∂A˜u∂ x= 0,

∫ xN

0A(x, t)dx=

bα + 1

, (3.2a,b)

where ˜u(x, t) is the average of u(x, y, z, t) over the area A. The challenge is to findu that satisfies (3.1) and the abovementioned boundary conditions; it should be noted,however, that the right-hand side of (3.1) is a function of x, t.

For propagation and interface, we seek a self-similar solution of the type

xN =KV tβVh, h= (xN)−1/(α+1)H(ξ), ˜u= βVhKV tβVh−1U(ξ), (3.3a−c)

with ξ = x/xN ∈ [0, 1]. Here, the subscript V means viscous, to distinguish from theself-similar solution for the inertial (inviscid) flow considered in § 2, and h stands for‘homogeneous’.

Takagi & Huppert (2007) obtained analytical solutions for V-shaped and semicircularsections, α = 1, 1/2).

For the generic power-law cross-section, approximations in the spirit of the boxmodel, or order-of-magnitude arguments, are useful and are reported in the following.The ratio of gap to height, bhα−1, depends strongly on α when h→ 0. Therefore, wedistinguish between the cases.

(i) In the ‘wide’ section, α < 1, the z shear (second term in (3.1)) is dominant; (ii)in the ‘narrow’ section, α > 1, the y shear is dominant. Therefore, (3.1) yields thefollowing order-of-magnitude balances:

˜u∼−(Reh0

x0

)h 2 ∂ h∂ x∼

(Re

h0

x0

)h 3

xN(α < 1), (3.4)

˜u∼−(Reh0

x0

)∂ h∂ x∼

(Re

h0

x0

)h2α+1

xN(α > 1). (3.5)

Using (3.3) (with H = 1 and U = 1 for definiteness), we find that these balancesproduce

βVh =α + 12α + 5

for α 6 1, βVh =α + 14α + 3

for α > 1, (3.6a,b)

and for any α > 0,

βVhK1/βVhV ∼

(Re

h0

x0

). (3.7)

In general, since h = hN = 0 at xN , the height function must display the singularbehaviour H(ξ)= c(1− ξ)p for ξ→ 1. The value of p is determined by the conditionthat uN > 0 in (3.4)–(3.5); we obtain p = 1/3, 1/(2α + 1) for α 6 1 and α > 1respectively.

A comparison of the rigorous solutions of Takagi & Huppert (2007) for a∨

triangle and semicircle (α= 1, 1/2) with the above approximate derivation shows fullagreement concerning βVh and p, and fair agreement for KV .

For α = 0 (wide rectangle, b> 1), we obtain the standard value βVh = 1/5, whichturns out to be the smallest value. For α = 1 (triangle), we obtain the largest valueβVh = 2/7.

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Gravity currents in critical condition 589

3.2. Current of variable volume advancing into a homogeneous (S= 0) or linearlystratified ambient (S= 1)

The derivations of βVh and xV in § 3.1 can be extended to currents of volume V = qtδ.We start with changes in (3.2b) (replace right-hand side with q t δ) and (3.3b) (replacexN with xN/(q tδ)). The details are not pursued here.

We first consider a ‘wide’ cross-section (α 6 1) and a homogeneous ambient fluid,S= 0. Let

xN =KV tβVh, h=(

q tδ

xN

)1/(α+1)

H(ξ), ˜u= βVhKV tβVh−1U(ξ). (3.8a−c)

Substitution into (3.4) yields for α 6 1

βVh =α + 1+ 3δ

5+ 2α, (3.9)

KV ∼

(Re

h0

x0

)(α+1)/(2α+5)

q3/(2α+5). (3.10)

For a current advancing in a linearly stratified ambient fluid with S = 1, the timeexponent in the viscous regime for ‘wide’ cross-sections is similarly derived as

βVs =α + 1+ 4δ

6+ 2αforα 6 1. (3.11)

Similar scalings are derived for ‘narrow’ cross-sections (α > 1) in both thehomogeneous (S = 0) and linearly stratified (S = 1) cases. The relevant exponentsβVh and βVs are reported in table 1, which includes also the scalings obtained in § 2for inertial flows.

4. Transition lengthNext, we proceed to the evaluation of xV where transition between inertial and

viscous regimes occurs. We observe that the typical transition is between self-similarpropagation of the form xN (t) = Kjtβjh , with j = V for viscous flow and j = I forinertial flow. We argue that the transition is smooth, and hence at this occurrence bothregimes display the same speed βjhKjt βjh−1 and the same length xV . Elimination oft= (xN/Kj)

1/βjh yields, after some algebra,

xVh =

(βVhK1/βVh

V

βIhK1/βIhI

)λ, λ=

βIhβVh

βIh − βVh. (4.1a,b)

The values of KI and βIh for the inertial regime are available (see § 2); βVh is givenby (3.6). Precise values of KV are available for some cases (α = 0, 1/2, 1), and ingeneral the approximation (3.7) can be used. The general result is

xVh = c(

Reh0

x0

)λ, λ=

βIhβVh

βIh − βVh, (4.2a,b)

where c is of order unity (the precise value depends on KV and can be calculated forα = 0, 1/2, 1). The result (4.2) can be extended to a linearly stratified ambient fluid

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590 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

2

1

2

3

4

0 3 4 51

5

6

FIGURE 2. (Colour online) The power λ for xV ∼ (Re(h0/x0))λ versus α for homogeneous

ambient fluid (S= 0) for lock release, δ= 0 (continuous thin line), waning, δ= 0.5 (dashedthin line), and constant influx, δ=1 (dotted thin line). The symbols refer to the rectangular(λ= 2/7), semicircular (λ= 3/8) and triangular (λ= 4/9) cross-sections. The thick linesrefer to a linearly stratified ambient fluid with maximum stratification (S= 1).

and S= 1. The value of the exponent λ can be evaluated from the time exponents βIh,βIs, βVh and βVs reported in table 1. Figure 2 shows the exponent λ for homogeneous(thin lines) and linearly stratified (thick lines) ambient fluid, for three different valuesof δ. We obtained a significant generalization of the standard transition-length formula(see Ungarish 2009 § 2.7): xV (scaled with x0 of the lock) depends on (Reh0/x0) atsome power λ ∈ [0.29, 0.44] for a variety of geometries of the cross-section. Theshortest transition for lock release is for the standard wide rectangle, λ= 2/7≈ 0.29;the longest is for the

∨triangle, λ= 4/9≈ 0.44. For the semicircle, λ= 3/8= 0.375.

For lock release in a linearly stratified ambient fluid with S = 1, the exponent issmaller with respect to the corresponding case in homogeneous ambient fluid, beingλ= 2/9≈ 0.22 for the rectangle, λ= 3/10= 0.3 for the semicircle and λ= 4/11≈ 0.36for the

∨triangle.

We must keep in mind, however, that (4.2) is just an estimate. Transition betweenregimes occurs over a distance rather than at a certain xV , and a careful calculationof the coefficient c in (4.2) is needed for rigour.

The xV estimate was subjected to laboratory tests for a triangle by Ungarish,Mériaux & Kurz-Besson (2014) (see figure 6 there), and for a semicircle byLongo et al. (2015b) (see figure 19 there). In general, the theoretical equation (4.2)underpredicts the measured xV . However, there is flexibility in the definition of theexperimental xV , and hence more comparisons are needed for a sharper conclusion.

5. Critical regime

In this section, we consider gravity currents with volume given by qtδ undera critical regime: at some critical δc, dependent of α (channel shape), we obtainβVh = βIh = βc, and the regime (inertial or viscous) is preserved during propagation

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Gravity currents in critical condition 591

(it should be noted that, in this case, λ= 0 and xVh =∞; see (4.2)). This behaviourfor the standard 2-D current is well documented (see Ungarish 2009 § 4.2.3), but theextension to other f (z)= bzα cases is as of yet unavailable.

5.1. Homogeneous ambient fluid, S= 0, ‘wide’ sectionWe first consider a current propagating in a homogeneous ambient fluid, S = 0. Forsmall δ, we expect that the viscous βVh will be smaller than the inviscid βIh, and forsome critical δc, we expect βIh= βVh. Let us estimate the critical point for a geometrybzα, α 6 1.

We consider the viscous current, as described in § 3. Let

xN =KV tβVh, h=(

q tδ

xN

)1/(α+1)

H(ξ), ˜u= βVhKV tβVh−1U(ξ). (5.1a−c)

Substitution into (3.4) yields

βVh =α + 1+ 3δ

5+ 2αfor α 6 1, (5.2)

KV ∼

(Re

h0

x0

)(α+1)/(2α+5)

q3/(2α+5). (5.3)

For the inviscid current, we use (2.18). Then, βIh = βVh yields the critical values

δc =2α + 7

4, βch =

54. (5.4a,b)

The 2-D rectangular counterpart corresponds to α= 0. The results agree with Ungarish(2009) § 4.2.3 (note the different notation). Interestingly, βch does not depend on thegeometry. For the semicircle α = 1/2, we obtain δc = 2.

5.2. Linearly stratified ambient fluid and S= 1, ‘wide’ sectionFor a current advancing in a linearly stratified ambient fluid and with S = 1, thetime exponents in the inertial and viscous regimes are reported in table 1. Again, thecondition βIs = βVs yields the critical value

δc =α + 4

2, βcs =

32. (5.5a,b)

As for the homogeneous ambient fluid, βcs does not depend on the shape of the cross-section. For the semicircle α = 1/2, we obtain δc = 9/4.

5.3. Generalization to ‘narrow’ sectionsThe extension to a ‘narrow’ power-law cross-section is straightforward. Table 2 liststhe theoretical values of δc and βc, also depicted in figure 3. The critical conditionalways requires a waxing current (δ > 1), with a maximum value δ for a

∨triangular

cross-section (α= 1) and with δc→ 3/2 for increasing α (very ‘narrow’ sections). Thecurrent is always accelerating (βc> 1); as α→∞, βc→ 1, i.e. the critical current in avery narrow fracture has a constant front speed. The values of the critical parametersδc and βc are larger for a stratified than for a homogeneous ambient fluid. The impactof stratification on the critical parameters is more pronounced for ‘wide’ than for‘narrow’ sections. The transition between α < 1 and α > 1 is sharp only in the presentalgebraic analysis which considers as dominant the shear stress in the horizontal(narrow section) or in the vertical plane (wide section), and is expected to be smoothin a more detailed study that includes the real distribution of the shear stress.

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592 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

2

1

2

3

0 3 4 51

FIGURE 3. (Colour online) Critical values of δ and β as functions of the shape parameterα. The thin lines refer to the homogeneous ambient fluid and the thick lines refer to alinearly stratified ambient fluid with maximum stratification (S= 1).

Ambient fluid ‘Wide’ section, ‘Narrow’ section,0 6 α 6 1 α > 1

δc βc δc βc

Homogeneous, S= 07+ 2α

454

3(2α + 1)4α

4α + 14α

Linearly stratified, S= 1α + 4

232

3α + 22α

2α + 12α

TABLE 2. Critical values of δ and β for ‘wide’ and ‘narrow’ cross-sections and for ahomogeneous (S= 0) and a linearly stratified ambient fluid (S= 1).

5.4. Scaling of the front position

In critical conditions, the front position scales with time with the same exponent β ≡βc for both the viscous–buoyancy and inertial–buoyancy regimes; as a consequence,observation of the asymptotic tβ behaviour of the current cannot identify the regime.Hence, we focus on the effect of q as the parameter that appears in the coefficientK of the front position, that most frequently varies in real system and that can beeasily changed during the experiments. Detailed computations of the scaling of thefront position with q are given in appendices B–E for a semicircular cross-section (α=1/2). In this case, when the ambient fluid is homogeneous (S= 0), then βIh = βVh ≡

βch = 5/4, δ ≡ δc = 2, and the front position at a given time is ∝ q1/2 in the viscous–buoyancy regime and ∝ q1/4 in the inertial–buoyancy regime (for a rectangular cross-section, Maxworthy (1983) found 3/5= 0.6 and 1/3≈ 0.33 respectively). For S= 1,then βIs = βVs ≡ βcs = 3/2, δ ≡ δc = 9/4 and the front position at a given time is ∝q4/7 in viscous–buoyancy balance and ∝ q2/5 in inertial–buoyancy balance respectively.

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Gravity currents in critical condition 593

S= 0

Viscous, xNV ∝

(q√

r

)1/2 (µc

ρcg′

)−1/4

t5/4

Inertial, xNI ∝

(q√

r

)1/4

Fr3/4g′3/8t5/4

δc = 2, βch = 5/4

S= 1

Viscous, xNV ∝

(q√

r

)4/7 (µc

ρ0N 2

)−3/14

t3/2

Inertial, xNI ∝

(q√

r

)2/5

N 3/5t3/2

δc = 9/4, βcs = 3/2

TABLE 3. The length of the current (dimensional) in a semicircular cross-section, forthe viscous–buoyancy and inertial–buoyancy regimes in critical conditions, for S = 0(homogeneous ambient fluid) and S= 1 (linearly stratified ambient fluid with a maximumstratification).

The resulting expressions are summarized in table 3 and are useful to interpret theexperimental tests of § 6.

Extension of the analysis to a generic power-law cross-section f (z)= bzα gives thefollowing for the inertial–buoyancy scaling.

For S= 0,

KIh =

[(α + 1)q

b

]1/(3+2α) (Fr√

g′

βch

)(2+2α)/(3+2α)

. (5.6)

For S= 1,

KIs =

[(α + 1)q

b

]1/(2+α) ( FrNβcs

√2

)(1+α)/(2+α). (5.7)

A generalization of the dependence of xN ∝ qχ for different cross-sections is listedin table 4 and shown in figure 4. The exponent χ is larger in the viscous–buoyancythan in the inertial–buoyancy regime, and the difference increases with α and is almostconstant for ‘wide’ sections. The theoretical difference of χ is sufficiently large forsuggesting that this is a good criterion for accurate identification of the regime ofpropagation in experiments with critical conditions and various q. This prediction istested and confirmed in § 6.

5.5. Estimate of the regime in critical conditionWe have established that a proper way to experimentally detect the regime (viscous–buoyancy or inertial–buoyancy) of a current in critical condition is to performexperiments with different values of q (all the other parameters are kept constant)and to observe the dependence of the front position on q.

In order to estimate the balance, we first consider a semicircular cross-section andthe case S = 0, with V = qt2 and βch = 5/4. The effective ratio between inertial andviscous forces is

Ree =uNh2

N

νxN. (5.8)

By using the inertial branch and the box-model approximation with h = hN , xN =

Ktβch→ uN =βxN/t, xN = uN =Fr√

g′hN , A= (4/3)√

2rh3/2, V =AxN , all in dimensional

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594 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

2

0.2

1.0

0 3 4 51

0.4

0.6

0.8

FIGURE 4. (Colour online) The dependence of the front position xN on q in criticalconditions, xN ∝ qχ , for S = 0 in the viscous (thin continuous line) and inertial (thindashed line) regimes, and for S=1 in the viscous (thick continuous line) and inertial (thickdashed line) regimes. The symbols indicate the values of χ for the present experiments ina semicircular cross-section (α = 1/2) with homogeneous (circles) and stratified (squares)ambient fluid.

Ambient fluid Viscous ‘wide’ Viscous ‘narrow’ Inertial,section, 0 6 α 6 1 section, α > 1 α > 0

Homogeneous q3/(5+2α) q(1+2α)/(3+4α) q1/(3+2α)

Linearly stratified, S= 1 q2/(3+α) q1/2 q1/(2+α)

TABLE 4. The dependence of the front position on q for ‘wide’ and ‘narrow’cross-sections.

form, substitution into (5.8) yields

Ree ≈75

64ν1

Fr√

2

q√

g′r. (5.9)

The balance is viscous–buoyancy if Ree is below a threshold value of order unity;otherwise it is inertial–buoyancy. For constant ν, r, the inertia/viscous forces ratiobehaves like q/

√g′.

Along the same lines, the effective Reynolds number for a current in a semicircularcross-section with S= 1, with V = qt9/4 and βcs = 3/2, is

Ree ≈3

2ν1

(1+ γ )2/5

(9

8Fr

)4/5 ( qN√

r

)4/5

. (5.10)

We can generalize the estimate of Ree for a cross-section f (z)= bzα, α6 1 (bhα isthe width, equal to 2

√2rh for the approximated circular cross-section).

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Gravity currents in critical condition 595

For S= 0, let l= 4/(2α + 3) (also equal to 1/(δch − 1)),

Ree =1νβ l+1

c

(α + 1

Frq

b√

g′

)l

. (5.11)

For S= 1, let p= 2/(α + 2) (also equal to 1/(δcs − 1)),

Ree =1νβ p+1

c1

(1+ γ )p/2

[(α + 1)

√2

FrqN b

]p

. (5.12)

In the ‘narrow’ channel, α > 1, the lateral shear is dominant and hence the effectiveRee is given by

Ree =uN(bhαN)

2

νxN. (5.13)

Using the same box-model estimates for xN, uN, hN , we obtain the following.For S= 0, let l= 4α/(2α + 3),

Ree =1νβ l+1

c b2−l

[α + 1

Frq√

g′

]l

. (5.14)

For S= 1, let p= 2α/(α + 2),

Ree =1νβ p+1

c b2−p 1(1+ γ )p/2

[(α + 1)

√2

FrqN

]p

. (5.15)

For the triangle, α = 1, the ‘wide’ and ‘narrow’ Ree coincide only for b= 1. Thisis because b= (width/height), and hence we should use (5.8) for b> 1 and (5.13) forb< 1.

The conclusion is that in the critical current system (influx with critical δc), theratio of inertia to viscous forces is proportional to q/

√g′ for S= 0, and to q/N for

S= 1. For a given current, this ratio may be large or small, but it is time-independent.The current is expected to be in the inertial regime if Ree is large and in theviscous regime if this parameter is small. It is convenient to define a threshold valueReet which separates the regimes. Some intermediary regime must exist, but theexperimental work of Maxworthy (1983) indicated that Reet is fairly sharp for a 2-Dcurrent, and we therefore expect a similar behaviour in the general case. Since boththe inviscid and the viscous theories lose accuracy in the intermediate regime, thereliable value of Reet must be determined by experiments, as detailed later.

6. Experimental layout and proceduresA series of experiments were planned and executed in the Hydraulic Laboratory

of the University of Parma to test the validity of the models developed for agravity current propagating into homogeneous and inhomogeneous ambient fluid incritical conditions. The experiments were conducted in a circular tube of polymethylmethacrylate (PMMA), a transparent thermoplastic, with an internal radius r= 9.5 cmand a length of 605 cm; see figure 5. The same apparatus was used by Longo et al.(2016b). A pipe connected to a vane pump was positioned in the inflow section. Thepump was equipped with an inverter for discharge feedback control; the sensor was

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596 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

Mirror

9.5 cm

Mirrors

605 cm

Collector

Video camera

Influx

x

Trigger signal

Inflow section

Valve

FIGURE 5. (Colour online) The layout of the experimental set-up.

a turbine meter operating in the flow rate interval 5–250 cm3 s−1, with an overallaccuracy equal to ±1 cm3 s−1. The inflow rate time dependence Q = δqtδ−1 wasimposed via software by a PC through a DAQ board. For inflow rates less than thethreshold of the turbine meter, the pump was controlled without a feedback system,with a voltage signal controlling the inverter proportional to the target inflow rate,according to a calibration curve.

In a first series of tests, the ambient fluid was homogeneous; the second seriesof tests required a linearly stratified ambient fluid, which was obtained with themethodology and the devices described in Longo et al. (2016a). To check the densityprofile in the tank, a small quantity of fluid was drained at different depths in thetank with a syringe connected to a small pipe of 0.1 cm in diameter. Then, themass density of the sample was measured by a hydrometer with an uncertainty of10−3 g cm−3. Figure 6 depicts the normalized density profiles for several experiments,which collapse on the theoretical line with a limited discrepancy.

The position of the front of the currents was determined using a full HD videocamera (Canon Legria HF 20; 1920× 1080 pixels) with a data rate of 25 frames persecond; the camera was translated parallel to the pipe in order to keep the nose of thecurrent in the field of view. A grid stuck at the bottom of the tube was used to detectthe front position of the current with an overall uncertainty of less than 0.2 cm.

Since the experiments required an accurate and repeatable measurement of thefront position in time (in other experiments, only the asymptotic trend of the front isrequired, which is much less sensitive to disturbances), special attention was devotedto obtaining uniform inlet (in order to avoid disturbances due to, e.g., differentdissipation rates of the intruding current) and outlet conditions.

The inlet pipe was made of rubber which could be easily deformed in order tovary the cross-sectional area. Figure 7 shows the front position in time for three testswith ρc = 1.060 g cm−3 and with different areas of the inlet cross-section, equal to≈4, 7 and 9 cm2. There are small differences among the profiles; further, while near

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Gravity currents in critical condition 597

0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

z (c

m)

FIGURE 6. (Colour online) Density profiles measured in the tank. The symbols refer todifferent experiments, the straight line indicates the perfect linearity.

101 102

t (s)

101

100

102

FIGURE 7. (Colour online) Front position for three experiments with ρc = 1.060 g cm−3,with different inlet cross-section areas, A ≈ 4 cm2 (crosses), A ≈ 7 cm2 (stars) and A ≈9 cm2 (triangles), with V = 0.2t2 cm3 (t in seconds). The continuous line represents thetheoretical asymptotic self-similar solution xN ∝ t5/4. The currents are in the critical regimeand viscous–buoyancy balance.

the entrance the current with the smallest inlet section was the fastest (due to a jet-like behaviour), asymptotically the current injected from the largest inlet cross-section(≈9 cm2) was slightly faster than the other two currents, as a consequence of minordissipation. The dissipation was essentially due to the expansion of the current. Havingchecked that the area of the inlet pipe did not affect the shape of the xN(t) function,all of the experiments were carried out with the inlet pipe having the maximum area.

The data shown in figure 7 also indicate the extension of the disturbances due tothe inlet section before approaching the self-similar regime; the length of adaptation is

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598 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

100 200 300 400

200

400

600

0

t (s)

FIGURE 8. (Colour online) Position of the front measured for two tests in identicalconditions (Exp. 37), linearly stratified ambient fluid and S= 1, critical regime (δ= 9/4),with the active outflow (star symbols) and the passive outflow (plus symbols). The curverepresents the theoretical self-similar solution in viscous–buoyancy balance; see appendixD, equation (D 13).

approximately 30 cm (say 3H). This length of adaptation depends on several factorslike the discharge rate and the cross-section area of the inlet pipe with respect to thecross-section area of the ambient fluid.

The outflow section was delimited by a weir with a crest at a fixed position inorder to guarantee a depth of H = 9.5 cm during all of the tests with homogeneousambient fluid. For the tests with density stratified ambient fluid, a drain with 12small-diameter pipes distributed over the cross-section was inserted into the outflowsection in order to avoid selective withdrawal. The pipes were connected to a valve,which was regulated during the experiment in order to keep a constant level in thechannel. The efficiency and necessity of this arrangement were verified by comparingtwo tests conducted under identical experimental conditions, but using two differentoutflow systems, the weir (passive) and the control valve (active); see figure 8. Agood overlap between the experimental results is evident for approximately half ofthe channel length; then, the effect of the accumulation of the denser fluid becomesevident, as the outflow over the crest of the weir drains only the lighter fluid. Thetheoretical solution, computed by assuming a viscous–buoyancy regime (see Longoet al. 2015a), is also shown in the figure. This solution overpredicts both datasets,highlighting the approximations intrinsic in the model, which in addition generallycause underestimation of the dissipation level. The choice of the minimum densityof the current was mainly based on the need to guarantee an adequate accuracy inmeasurements and a reasonable correspondence of the experimental layout with themodel, with a high signal/noise ratio. In fact, the density of the fluid can be measuredwith very accurate instruments; hence, it was possible to handle experiments withg′ values of a few cm s−2. However, it is difficult to guarantee the homogeneityof the ambient fluid in the whole channel with a level of accuracy comparable tothe accuracy in measuring the density, since temperature variations or other external

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Gravity currents in critical condition 599

effects are a source of disturbances adding noise of comparable level to the signal.For this reason, we chose a minimum mass density equal to 1.030 g cm−3. Themaximum density is dictated by the need to satisfy the Boussinesq approximation,with ε < 10 %.

The theory assumes that the current is thin and the return flow is small. Theseconditions were satisfied in our experiments, with the exception of a small initial time.This was verified by visual inspection and also by calculations following the injectedvolume. In particular, it should be noted that in our experiments, the typical volumeof the initial ambient fluid was approximately 0.5πr2ltank = 85 l, while the volume ofthe current qtδmax was approximately 2–6 l. At tmax, the current was spread over thelength ltank with a typical thickness of 1–2 cm.

6.1. Uncertainty in variables and parametersThe relative uncertainty in g′ was 0.2 % and the uncertainty in the stratificationparameter S = (ρb − ρ0)/(ρc − ρ0) was 1S/S = 3.2 %. The level of the ambientfluid was fixed with an accuracy of 0.1 cm, inducing a relative uncertainty of1H/H 6 1.1 %. The velocity scale had an uncertainty of 1U/U 6 0.6 % and the timescale had an uncertainty equal to 1T/T 6 1.7 %. The influx rate was measured withan uncertainty equal to 1 % of the instantaneous value, and the dimensionless influxrate had an uncertainty equal to 1q/q 6 4.3 %. By assuming an uncertainty of 1 %in the value of the kinematic viscosity of the denser fluid, the resulting uncertaintyin the Reynolds number was 1Re/Re 6 2.7 %.

6.2. ExperimentsTwo series of experiments were performed, the first with homogeneous ambient fluid(22 experiments with varying q and three different density values of the current) andthe second with a linearly stratified ambient fluid and S = 1 (nine experiments withvarying q). The main parameters of the experiments are listed in tables 5 and 6.

Figures 9 and 10 show the front position of the currents for the homogeneousambient fluid and the linearly stratified ambient fluid (S = 1). The straight linesindicate the theoretical asymptotic regime of the self-similar solutions. For bothdatasets, there is a reasonable collapse of the data in the asymptotic limit; theexperimental β is slightly smaller than the theoretical one. The different value ofβ can be attributed in general to the dissipation level, which in the experiments isinvariably greater than the theoretical one. There are some other considerations. Forthe inertial current, in practice, Fr is not a constant as assumed. In the critical flow,h∼ t1/2 and Fr decreases; therefore, the practical propagation is slower, with a smallerβ. In the viscous current, there is shear from the side, which we neglected. This willalso hinder propagation and cause a smaller β. We also keep in mind that the circlesection is not a perfect power-law profile. This is also expected to contribute to thediscrepancy between theory and experimental data.

In order to detect the different regimes, we have fitted the experimental dataset foreach experiment with a function xN = a+K(q)tβc (see figure 10(b) for an example ofthe data-fit lines), obtaining a series of values K(q), shown in figure 11 for the fourseries of experiments. We expect that K(q)∝ qχ , where the theoretical χ is given intable 3 for the different regimes. In each series of data, it is possible to detect twobranches with an exponent slightly different from the theoretical value but showing asignificant variation in the two regimes. The vertical dashed lines indicate the valueof q that separates currents in a viscous–buoyancy regime (below) and currents in an

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600 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

Exp

t.δ

q(c

m3

s−2 )

ε(%)

g′(c

ms−

2 )R

e(×

103 )

U(c

ms−

1 )T(s)

a(c

m)

K(q)(c

ms−

5/4 )

Re e

Bal

ance

142

0.05

03.

4133

.514

.117

.80.

533

5.3

0.36

40.

14v–

b15

20.

101

3.41

33.5

14.1

17.8

0.53

33.

20.

502

0.28

v–b

162

0.15

03.

4133

.514

.117

.80.

533

2.4

0.59

70.

41v–

b17

20.

200

3.41

33.5

14.1

17.8

0.53

35.

60.

661

0.55

i–b

182

0.25

03.

4133

.514

.117

.80.

533

7.5

0.69

20.

68i–

b19

20.

299

3.41

33.5

14.1

17.8

0.53

32.

20.

722

0.82

i–b

202

0.34

93.

4133

.514

.117

.80.

533

3.9

0.76

50.

96i–

b6

20.

050

6.40

63.0

20.0

24.5

0.38

86.

40.

390

0.10

v–b

42

0.10

06.

4063

.019

.523

.90.

398

5.6

0.53

20.

21v–

b8

20.

150

6.40

63.0

20.0

24.5

0.38

83.

90.

701

0.31

v–b

422

0.29

96.

4063

.019

.324

.50.

388

1.8

0.88

00.

60i–

b43

20.

399

6.40

63.0

19.3

24.5

0.38

86.

60.

928

0.80

i–b

442

0.49

86.

4063

.019

.324

.50.

388

3.6

0.98

10.

99i–

b45

20.

598

6.40

63.0

19.3

24.5

0.38

89.

11.

025

1.19

i–b

462

0.69

56.

4063

.019

.324

.50.

388

3.3

1.07

51.

39i–

b28

20.

050

9.44

92.5

23.4

29.7

0.32

05.

50.

451

0.08

v–b

292

0.10

09.

4492

.523

.429

.70.

320

4.4

0.63

80.

16v–

b30

20.

150

9.44

92.5

23.4

29.7

0.32

03.

70.

777

0.25

v–b

312

0.20

09.

4492

.523

.429

.70.

320

6.4

0.90

20.

33v–

b32

20.

250

9.44

92.5

23.4

29.7

0.32

05.

20.

999

0.41

v–b

332

0.29

99.

4492

.523

.429

.70.

320

4.2

1.06

20.

49i–

b35

20.

347

9.55

93.6

23.6

29.8

0.31

97.

11.

110

0.50

i–b

TAB

LE

5.Pa

ram

eter

sof

the

expe

rim

ents

with

hom

ogen

eous

ambi

ent

fluid

(S=

0).

The

chan

nel

isci

rcul

arw

ithra

dius

r=9.

5cm

,ha

lffil

led

with

ambi

ent

fluid

(H=

9.5

cm).

The

volu

me

isV=

qtδ,ε=(ρ

c−ρ

0)/ρ

0,g′=

gεis

the

redu

ced

grav

ity,

U=√

g′H

isth

eve

loci

tysc

ale,

T=

H/U

isth

etim

esc

ale,

Re=

UH/ν

,w

hereν

isth

eki

nem

atic

visc

osity

ofth

ecu

rren

t.T

hete

rms

aan

dK(q)

are

the

coef

ficie

nts

ofth

ein

terp

olat

ing

curv

ex N=

a+

K(q)tβ

c,

with

unce

rtai

nty6

4%

and6

3%

ona

and

K(q)

resp

ectiv

ely;

Re e

isth

eef

fect

ive

Rey

nold

snu

mbe

r;th

eba

lanc

eis

visc

ous–

buoy

ancy

(v–b

)or

iner

tial–

buoy

ancy

(i–b

).

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Gravity currents in critical condition 601

Exp

t.δ

q(c

m3

s−9/

4 )ε(%)

g′(c

ms−

2 )S

N(s−

1 )R

e(×

103 )

U(c

ms−

1 )T(s)

a(c

m)

K(q)(c

ms−

3/2 )

Re e

Bal

ance

372.

250.

023

8.18

80.2

1.0

2.90

21.8

27.6

0.34

45.

30.

093

0.79

v–b

382.

250.

035

8.08

79.2

1.0

2.89

21.7

27.4

0.34

63.

40.

127

1.09

v–b

352.

250.

044

8.22

80.6

1.0

2.91

21.9

27.7

0.34

35.

40.

138

1.33

v–b

392.

250.

063

8.39

82.3

1.0

2.94

22.1

28.0

0.34

04.

20.

188

1.77

v–b

472.

250.

080

8.17

80.1

1.0

2.90

21.8

27.6

0.34

411

0.21

62.

08v–

b48

2.25

0.08

78.

1580

.01.

02.

9021

.827

.60.

345

3.3

0.21

92.

22v–

b49

2.25

0.14

08.

1880

.21.

02.

9021

.827

.60.

344

120.

306

3.24

i–b

502.

250.

188

8.08

79.2

1.0

2.89

21.7

27.4

0.34

69.

80.

351

4.12

i–b

512.

250.

242

8.06

79.0

1.0

2.88

21.6

27.4

0.34

78.

20.

405

5.02

i–b

TAB

LE

6.Pa

ram

eter

sof

the

expe

rim

ents

with

linea

rly

stra

tified

ambi

ent

fluid

and

S=

1.T

hete

rmN

isth

ebu

oyan

cyfr

eque

ncy.

For

the

othe

rsy

mbo

ls,

see

the

capt

ion

tota

ble

5.

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602 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

(a) (b)Test

14151617181920

0.050.100.150.200.250.300.35

Test

6484243444546

0.050.100.150.300.400.500.600.70

t (s) t (s)

101

101

102

102 101 102

FIGURE 9. (Colour online) Front position for (a) seven tests with ρc = 1.030 g cm−3

and (b) eight tests with ρc= 1.060 g cm−3, with homogeneous ambient fluid (S= 0). Thestraight dashed lines indicate the front position, xN ∝ t5/4.

inertial–buoyancy regime (above). See also tables 5 and 6, containing the indicationof the balance and the effective Reynolds number.

For currents advancing in a homogeneous ambient fluid, the threshold separating thetwo regimes has been computed using the intersection point of the data-fit functionsof the lower and the upper series of experimental data in figure 11(a–c), where theintersection is observed for q≈ 0.17, 0.20, 0.27 cm3 s−2 for the experiments with ρc=

1.030, 1.060, 1.090 g cm−3 (ε = 3.4, 6.4, 9.4 %).By adopting Fr=

√2 in (5.9), the corresponding effective Reynolds number marking

the threshold is Reet = 0.46, 0.40, 0.45 for the three series, a value that is fairlyconstant. In comparing theory and experiments, numerous approximations inherent inthe model should be kept in mind: a parabolic cross-section in lieu of a circularone, a negligible effect of the ambient fluid dynamics, use of the box model, whichexcludes all the details of the current shape, and the fact that regime transition is nota sharp effect. Consequently, the fact that the threshold separating the two regimesoccurs at Ree approximately equal to 1 should be considered to be fair agreement withprediction. We can compare the present results for currents advancing in homogeneousambient fluid with similar results obtained by Maxworthy (1983) in a rectangularcross-section. For a rectangular cross-section with homogeneous ambient fluid, theeffective Reynolds number is

Ree =5

(5q

4Fr b√

g′

)4/3

. (6.1)

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Gravity currents in critical condition 603

28293031323335

0.050.100.150.200.250.300.35

373835394748495051

0.0230.0350.0440.0630.0800.0870.1400.1880.242

t (s) t (s)

Test Test(a) (b)

101

101

102

102 101 102

FIGURE 10. (Colour online) Front position for (a) seven tests with ρc = 1.090 g cm−3,with homogeneous ambient fluid (S = 0), and (b) nine tests with a linearly stratifiedambient fluid, ρc = 1.086 g cm−3, S = 1, and with data-fit curves of the equation xN =

a + K(q)tβc . The straight dashed lines indicate the front position, xN ∝ t5/4 and xN ∝ t3/2

respectively.

Inserting the data of the transition documented by Maxworthy (1983) in his threesets of experiments the effective threshold Reynolds number is Reet = 0.96, 0.89, 0.92,again of O(1). We remind readers that the experiments where transition takes place(Maxworthy 1983) are characterized by the following parameters: J = 2.9, 3.5, 3.4,q/b = 0.033, 0.030, 0.055 cm2 s−7/4, g′ ≡ (1ρ/ρ)g = 1.27, 1.18, 3.73 cm s−2, ν =0.01 cm2 s−1, [(KI(7/4)/KV(7/4))]15

= 2.2, where KI and KV are constants.For currents advancing in a linearly stratified ambient fluid (S = 1, figure 11d),

the intersection occurs at q≈ 0.13 cm3 s−9/4, with a corresponding effective thresholdReynolds number Reet ≈ 3 for Fr=

√2 in (5.10).

7. Summary and conclusionsWe have investigated the behaviour of critical gravity currents, advancing with

both viscous–buoyancy and inertial–buoyancy regimes in a homogeneous ambientfluid (S = 0) and a linearly stratified ambient fluid with maximum stratification(S = 1). In both regimes, a self-similar solution is expected. The analysis has beendetailed theoretically for a generic power-law cross-section, f (z) = bzα, and checkedexperimentally for a semicircular cross-section, locally approximated by a parabola(α = 1/2). It is important to keep in mind the behaviour of the current in differentconditions.

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604 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

(a) (b)

(c) (d)

10–1

10–2 10–1 100

100

10–1

10–1

100

100

10–1

100

10–2

10–2

10–1 100

10–2 10–1 10010–2 10–1 100

FIGURE 11. (Colour online) The coefficient K(q) of the front position xN = K(q)tβc .(a–c) Homogeneous ambient fluid with ρc = 1.030, 1.060, 1.090 g cm−3 respectively; thetheoretical dependence is K(q) ∝ q1/2 in viscous–buoyancy balance and K(q) ∝ q1/4 ininertial–buoyancy balance. (d) Linearly stratified ambient fluid, ρc = 1.086 g cm−3, S= 1;the theoretical dependence is K(q) ∝ q4/7 in viscous–buoyancy balance and K(q) ∝ q2/5

in inertial–buoyancy balance. The dashed lines limit the 95 % confidence band, and thevertical dashed lines indicate the value of q separating the viscous–buoyancy (below) andinertial–buoyancy (above) regimes.

(1) In subcritical condition (δ < δc), the current starts in the inertial–buoyancy regimeand then at a finite time turns into the viscous–buoyancy state, with a smaller β.The value of Ree is initially large and decreases with time (and xN).

(2) In supercritical condition (δ > δc), the current starts in the viscous–buoyancyregime and then (again at a finite time) turns into the inertial–buoyancy state,

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Gravity currents in critical condition 605

with a smaller β. The value of Ree is initially small and increases with time(and xN).

(3) In critical condition, the current is in the inertial or the viscous regime from thebeginning. Both regimes have the same β. The current maintains the regime (i.e.Ree is constant during the propagation).

Our work leads to the following major conclusions.

(1) A collection of relatively simple closed-form solutions is available to describe theadvancement of GCs in inertial and viscous regimes in a fairly realistic set-upincluding a channel of assigned shape and a stratified ambient. Self-similarity issatisfied within the experimental uncertainty.

(2) Specific scalings are derived for the propagation of the current in criticalconditions. A stratified ambient requires a larger δ and a larger accelerationto reach the critical condition as compared to a homogeneous ambient. Theimpact of stratification on the critical parameters is more pronounced for ‘wide’than for ‘narrow’ sections.

(3) A subcritical current (δ < δc) starts in inertial–buoyancy balance and then turnsto viscous–buoyancy balance; a supercritical current (δ > δc) starts in viscous–buoyancy balance and then turns to inertial–buoyancy balance.

(4) A critical current, after an initial adaptation, propagates according to theasymptotic represented by the self-similar solution for the specific regimedictated by the effective Reynolds number: viscous–buoyancy if Ree < Reet;inertial–buoyancy if Ree > Reet. However, the precise value of the effectiveReynolds number separating the two regimes cannot be evaluated from ourmodel, and must be obtained empirically.

(5) Since the asymptotic xN ∼ tβ behaviour of the current in critical condition cannotbe used to discriminate the regime (viscous–buoyancy or inertial–buoyancy), thedependence of the front position as a function of the discharge parameter qcan be used for this purpose. The experimental results for a current with threedifferent values of g′, advancing in a homogeneous ambient fluid, show that theflow is in a viscous–buoyancy balance below a threshold value of q and is inan inertial–buoyancy balance above this value, with good agreement with theorywithin the uncertainties.

(6) For a semicircle, we obtained the experimental effective threshold Reynoldsnumber separating the two different balances, Reet, is ≈0.5 for a homogeneousambient fluid and ≈3 for a linearly stratified one, consistent with theoreticalpredictions of Reet = O(1). Figure 12 contains the key elements of theanalysis, which has been developed for the generic cross-section and has beenexperimentally validated for the circular cross-section.

The laboratory preparations and observations provided specific insights into themechanics of the current and suggestions about the management of the experiments,as follows.

(1) Preliminary tests show that the inlet conditions, varied by changing the area ofthe inlet pipe, have a minor effect on the propagation of the currents, whosespeed is only marginally affected by the extent of the dissipation at the inlet.This loss of memory with respect to the initial and boundary conditions wasalready known for GCs, but had never been tested in experiments with variable

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606 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

(See table 2)

ItV

VtI

See table 1y

z

y

z

q

1

2

2a

2b

3

4

FIGURE 12. (Colour online) Diagram of the theoretical propagation-behaviour of a gravitycurrent in a power-law channel of width f (z)= bzα with homogeneous, S= 0, or stratified,S= 1, ambient and volume V = qtδ .

inflow rate. A significant effect of ‘selective withdrawal’ was identified forGCs advancing in a linearly density stratified fluid. This effect significantlyslows down the current; hence, care must be taken to avoid it in setting up theexperiments.

(2) The experimental value of β is slightly lower than the theoretical value as aconsequence of numerous effects neglected by the simple model adopted forcomparison.

Future work could include a better modelling of the advancing current via atwo-layer model, and more detailed experiments to check the transition length forsubcritical and supercritical currents.

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Gravity currents in critical condition 607

AcknowledgementsWe thank I. Lauriola for her support during the revision of the paper.

Appendix A. Numerical verification of the asymptotic regimeIn order to check that the asymptotic behaviour of the current evolves towards the

self-similar solution represented by (2.16), we have integrated the equations (2.23)–(2.24) valid for homogeneous ambient fluid (S= 0) with the method of characteristics.The (time-varying) integration domain was mapped onto [0, 1] through the transforms(2.15) and then the equations written in the new domain were integrated along thetwo trajectories with space step and time step chosen in order to guarantee a Courantnumber ≈ 0.5, usually equal to 1/100 and 1/200 respectively. The boundary conditionat the nose is given by (2.9) and the boundary at the inlet follows the indications given

in Shringarpure et al. (2013), with a fixed inlet Frin= uin/

√hin, where Frin is the inlet

Froude number. The inlet velocity uin and current depth hin are time varying in orderto satisfy the time-varying discharge and the constant inlet Froude number.

Figure 13(a) shows the front position as a result of the numerical integration fora circular cross-section and δ = 2, and figure 13(b) shows the time variation of theexponent computed as β = ˜xN t/xN , which tends to assume the theoretical value β ≡βIh = 5/4. Figure 13(c) shows the current depth at different times.

The numerical integration was also performed for the equations (2.25)–(2.26) validfor stratified ambient fluid (S= 1). Figure 14(a) shows the front position as a resultof the numerical integration for a circular cross-section and δ= 9/4, and figure 14(b)shows the time variation of the exponent, which tends to assume the theoretical valueβ ≡ βIs = 3/2. Figure 14(c) shows the current depth at different times.

Appendix B. Viscous–buoyancy regime, S= 0

For a semicircular cross-section, the viscous–buoyancy force balance gives thefollowing scales (see Longo et al. 2015a):

x∗ =(

q√

r

)2/(2δ+5)(µc

ρcg′

)2δ/(2δ+5)

, (B 1)

t∗ =(

q√

r

)−2/(2δ+5)(µc

ρcg′

)5/(2δ+5)

, (B 2)

where g′ = (ρc − ρ0)/ρcg is the reduced gravity, and the dimensional length of thecurrent is

xNV = ηN(δ)

(KC

2√

2

)1/2 ( q√

r

)(2βVh+2)/(2δ+5) (µc

ρcg′

)(2δ−5βVh)/(2δ+5)

tβVh . (B 3)

In critical condition (δ ≡ δch = 2, βVh ≡ βch = 5/4), (B 3) yields

xNV = ηN|δ=2

(KC

2√

2

)1/2 ( q√

r

)1/2 (µc

ρcg′

)−1/4

t5/4, (B 4)

where ηN|δ=2 ≈ 1.4 and KC = 32√

2/105≈ 0.43.

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608 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

0

1

2

3

4

(a) (b)

(c)

0.6

0.8

1.0

1.2

1.4

0.4 0.8 1.2 1.6 2.0 0 0.4 0.8 1.2 1.6 2.0

0 1 2 3 4

0.2

0.4

0.6

0.8

1.0

FIGURE 13. (Colour online) Numerical model results for a current advancing in a circularcross-section in the inertial–buoyancy regime in a homogeneous ambient fluid (S = 0),with volume ∝ t2 (δ = 2). (a) The time series of the front position xN (symbols) andthe interpolating function of the equation xN ∝ t1.24 (continuous red curve); (b) the timeseries of the exponent βIh, with the dashed horizontal line representing the theoreticalvalue βIh = 5/4; (c) profiles of the current at different times.

Appendix C. Inertial–buoyancy regime, S= 0

The dimensional expression of the length of the current in the inertial regime iscomputed by evaluating the length and time scales. The front condition reads as(dimensional)

xN = Fr√

g′hN, (C 1)

which yields

u∗˜xN = Fr√

g′x∗

√hN, (C 2)

where the tilde indicates that the variable is dimensionless; u∗ and x∗ are the velocityand length scales respectively. The dimensional homogeneity requires u∗ = Fr

√g′x∗.

The integral mass conservation is

4√

2r3

∫ xN (t)

0

h3/2 dx= qtδ (C 3)

or4√

2r3

x5/2∗

∫ xN (t)

0

h3/2 dx= qtδ∗tδ; (C 4)

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Gravity currents in critical condition 609

0

0.4

0.8

1.2

0.8

1.2

1.6

0.4 0.8 1.2 0 0.4 0.8 1.2

0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

(a) (b)

(c)

FIGURE 14. (Colour online) Numerical model results for a current advancing in a circularcross-section in the inertial–buoyancy regime in a linearly stratified ambient fluid (S= 1),with volume ∝ t9/4 (δ = 9/4). (a) The time series of the front position xN (symbols) andthe interpolating function of the equation xN ∝ t1.47 (continuous red curve); (b) the timeseries of the exponent βIs, with the dashed horizontal line representing the theoretical valueβIs = 3/2; (c) profiles of the current at different times.

hence,√

rx5/2∗= qtδ

∗. By solving the system of equations (where u∗= x∗/t∗), the scales

are equal to

x∗ =(

q√

r

)2/(5−δ) ( 1Fr√

g′

)2δ/(5−δ)

, (C 5)

t∗ =(

q√

r

)1/(5−δ) ( 1Fr√

g′

)5/(5−δ)

. (C 6)

Hence,

xNI ∝

(q√

r

)(2−βIh)/(5−δ) ( 1Fr√

g′

)(2δ−5βIh)/(5−δ)

tβIh . (C 7)

In critical condition (δ ≡ δch = 2, βIh ≡ βch = 5/4), (C 7) yields

xNI ∝

(q√

r

)1/4

Fr3/4g′3/8t5/4. (C 8)

Appendix D. Viscous–buoyancy regime, S= 1

Use of (2.5) gives∂p∂x= ρ0 N 2h

∂h∂x. (D 1)

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610 L. Chiapponi, M. Ungarish, S. Longo, V. Di Federico and F. Addona

For the viscous controlled regime, the force balance FV ∼ FB is required. In acircular cross-section approximated by a parabola (α = 1/2), the dominant shearstress acts along the vertical; hence,

µc∂2u∂z2=−ρ0 N 2h

∂h∂x, (D 2)

which in turn allows the computation of the velocity u(x, y, z) by imposing theboundary conditions at the wall of the cross-section and at the interface with theambient fluid.

Integration of the velocity over the cross-section of the current gives the volumedischarge as

Q= h7/2√rKCρ0

µcN 2h

∂h∂x, (D 3)

where KC = 32√

2/105 is a coefficient, with the subscript standing for ‘circular’. Thedifferential form of the mass conservation reads as

h1/2 ∂h∂t−

√2KC

4ρ0 N 2

µc

∂x

(h9/2 ∂h

∂x

)= 0, (D 4)

which, by introducing the length and the time scales, can be written as

x3/2∗

t∗h1/2 ∂ h

∂ t−

√2KC

4ρ0 N 2

µcx7/2∗

∂ x

(h9/2 ∂ h

∂ x

)= 0. (D 5)

The integral mass conservation is again (C 4).The corresponding length and time scales are computed from (D 5) and (C 4) as

x∗ =(

q√

r

)2/(4δ+5) (µc

ρ0 N 2

)2δ/(4δ+5)

, (D 6)

t∗ =(

q√

r

)−4/(4δ+5) (µc

ρ0 N 2

)5/(4δ+5)

. (D 7)

The system of equations (D 5)–(C 4) then becomes, in dimensionless form,

h1/2 ∂ h∂ t−∂

∂ x

(h9/2 ∂ h

∂ x

)= 0,

4√

23

∫ xN (t)

0h3/2 dx= t δ, (D 8a,b)

and admits a self-similar solution expressed as

h= η1/2N t (2δ−1)/7f (η), η=

(2√

2KC

)1/2

x t−(8δ+3)/14, (D 9a,b)

where

ηN = ηN(δ)=

[4√

23

(KC

2√

2

)1/2 ∫ 1

0f 3/2 dζ

]−4/7

ζ =η

ηN. (D 10)

The length of the current is then

xV = ηN

(KC

2√

2

)1/2

t βVs, (D 11)

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Gravity currents in critical condition 611

where βVs= (8δ+ 3)/14. Finally, the (dimensional) length of the current in the viscousregime for S= 1 is derived as

xNV = ηN

(KC

2√

2

)1/2 ( q√

r

)(2+4βVs)/(5+4δ) (µc

ρ0 N 2

)(2δ−5βVS)/(5+4δ)

tβVs . (D 12)

In critical condition (βVs ≡ βcs = 3/2, δ ≡ δcs = 9/4), (D 12) yields

xNV = ηN|δ=9/4

(KC

2√

2

)1/2 ( q√

r

)4/7 (µc

ρ0 N 2

)−3/14

t3/2, (D 13)

with ηN|δ=9/4 ≈ 1.056.

Appendix E. Inertial–buoyancy regime, S= 1

The inertial–buoyancy balance is

ρcu∂u∂x∼ ρ0 N 2h

∂h∂x, (E 1)

which becomes, in dimensionless form,

ρcu2∗

x∗u∂ u∂ x∼ ρ0 N 2x∗h

∂ h∂ x. (E 2)

The continuity equation is again (C 4),

4√

2r3

x5/2∗

∫ xN (t)

0

h3/2dx= qtδ∗tδ. (E 3)

In the Boussinesq approximation (ρ0/ρc ≈ 1), equations (C 4)–(E 2) yield thecharacteristic length and time scales as follows:

x∗ =(

q√

r

)2/5

N−2δ/5, (E 4)

t∗ = N−1. (E 5)

The (dimensional) length of the current in the inertial–buoyancy regime is then

xNI ∝

(q√

r

)2/5

N (βIs−2δ)/5tβIs, (E 6)

where βIs = (3+ 2δ)/5. In critical condition (βIs ≡ βcs = 3/2 and δ ≡ δcs = 9/4), (E 6)reads as

xNI ∝

(q√

r

)2/5

N 3/5t3/2. (E 7)

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