Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 17.5.2 - A numerical investigation of constant-volume non-Boussinesq density currents
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Title: 17.5.2 - A numerical investigation of constant-volume non-Boussinesq density currents Environmental and Geophysical Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Bonometti, T.
Ungarish, M.
Balachandar, S.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: density current
non-Boussinesq effects
Navier-Stokes simulation
shallow-water theory
 Notes
Abstract: The time-dependent behaviour of non-Boussinesq high-Reynolds-number density currents of density c, released from a lock of height h0 and length x0 into a ambient of height H and density a, is considered. We use two-dimensional Navier-Stokes simulations to cover a wide range of density ratio c/ a (for both “heavy”-bottom and “light”-top currents) and geometric ratios (H*=H/h0, l=x0/h0). To our knowledge, the ranges of parameters and times of propagation considered here were not covered in previous experimental or numerical studies. In the first part, we set the lock aspect ratio to l=18.75, and vary the density ratio 10-4< c/ a<104 and initial depth ratio 1£H*£50. The Navier-Stokes results are compared with predictions of a shallow-water model, in the regime of constant-speed (slumping) phase. Good agreement is observed in a large region of the parameter space ( c/ a; H*). The larger discrepancy is observed in the range of high-H* and low- c/ a for which the shallow-water model overpredicts the velocity of the current. Two possible reasons are suspected, namely the fluid motion in the ambient fluid which is not accounted for in the model, and the choice of the model for the front condition. In the second part, we set the initial depth ratio to H*=10, and vary the density ratio 10-2< c/ a<102 and lock aspect ratio 0.5£l £18.75. In particular, we derive novel insights on the influence of the lock aspect ratio l=x0/h0 on the shape and motion of the current in the slumping stage. It is shown that a critical value exists, lcrit; the dynamics of the current is significantly influenced by l if below lcrit. We present a simple analytical model which support the observation that for a light current the speed of propagation is proportional to l1/4 when l<lcrit.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00427
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Holding Location: University of Florida
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Resource Identifier: 1752-Bonometti-ICMF2010.pdf

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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


A numerical investigation of constant-volume non-Boussinesq density currents


Bonometti Thomas*, Ungarish Mariusl, Balachandar S.t

University de Toulouse, INPT, UPS, CNRS, Institut de M6canique des Fluides de Toulouse,
All6e Camille Soula, F-31400 Toulouse, France
'Department of Computer Science, Technion, Haifa, 32000, Israel
tDepartment of Mechanical and Aerospace Engineering, University of Florida, Gainesville FL32611, USA
thomas.bonometti@imft.fr, unga@cs.technion.ac.il, balals@ufl.edu



Keywords: Density current, non-Boussinesq effects, Navier-Stokes simulation, shallow-water theory



Abstract

The time-dependent behaviour of non-Boussinesq high-Reynolds-number density currents of density pc, released from
a lock of height ho and length xo into a ambient of height H and density Pa, is considered. We use two-dimensional
Navier-Stokes simulations to cover a wide range of density ratio pc/pa (for both "heavy"-bottom and "light"-top currents) and
geometric ratios (H*=H7,,,, A = .,, i,,i To our knowledge, the ranges of parameters and times of propagation considered here
were not covered in previous experimental or numerical studies. In the first part, we set the lock aspect ratio to A= 18.75, and
vary the density ratio 10-4 of a shallow-water model, in the regime of constant-speed (slumping) phase. Good agreement is observed in a large region of
the parameter space (pl/a; H*). The larger discrepancy is observed in the range of high-H* and low-pc/pa for which the
shallow-water model overpredicts the velocity of the current. Two possible reasons are suspected, namely the fluid motion in
the ambient fluid which is not accounted for in the model, and the choice of the model for the front condition. In the second part,
we set the initial depth ratio to H*=10, and vary the density ratio 10-2 particular, we derive novel insights on the influence of the lock aspect ratio 2= i,, i,, on the shape and motion of the current in the
slumping stage. It is shown that a critical value exists, Acr,; the dynamics of the current is significantly influenced by 2 if below
A2c,. We present a simple analytical model which support the observation that for a light current the speed of propagation is
proportional to 2A4 when A

Introduction

Constant-volume density currents have been studied
extensively because of their importance in various industrial
and environmental problems (e.g. Simpson 1982, Ungarish
2009). Horizontal density currents are buoyancy-driven
flows which manifest themselves as a current of heavy
(resp. light) fluid running below light (resp. above heavy)
fluid. Initially after release the current accelerates and
reaches a constant speed of spreading (referred to as the
slumping phase, Huppert & Simpson 1980). This phase
lasts until the backward propagating disturbance reflects off
the back wall or symmetry plane and propagates forward to
catch up with the front (Rottman & Simpson 1983). The
duration of the slumping phase depends on the Reynolds
number and volume of release. After the slumping phase,
the current velocity decreases in a self-similar manner at a
rate that depends on the dominant effect (inertia, viscosity,
surface tension).
In the Boussinesq limit, the dynamics of planar
currents of arbitrary initial depth ratios is relatively well
understood thanks to various laboratory experiments (e.g.


Rotmann & Simpson 1983; Marino, Thomas & Linden
2005), numerical investigations (Hartel, Meiburg & Necker
2000; Ozgokmen et al. 2004; Cantero et al. 2007, Ooi,
Constantinescu & Weber 2009) and analytical modelling
(Benjamin 1968; Huppert & Simpson 1980; Klemp,
Rotunno & Skamarock 1994). The dynamics of density
currents of arbitrary density ratios is less understood
(Ungarish 2009) and most of the previous reported work has
been restricted to the lock-exchange configuration at (i) a
fixed initial depth ratio (e.g. Lowe, Rottman & Linden
2005: Birman, Martin & Meiburg 2005; Etienne, Hopfinger
& Saramito 2005; Bonometti, Balachandar & Magnaudet
2008), or (ii) a fixed density ratio (Schoklitsch 1917; Martin
& Moyce 1952; Zukoski 1966; Gardner & Crow 1970;
Wilkinson 1982; Baines et al. 1985; Spicer & Havens 1985;
Lauber & Hager 1998; Stansby et al. 1998,). To our
knowledge, the only reported experimental work of
non-Boussinesq current for various initial depth ratios and
density ratios is that of Gribelbauer, Fannelop & Britter
(1993). Further understanding is of critical interest for the
prevention of hazardous situations such as fires in tunnels,






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


dam break, snow avalanche or accidental release of toxic
gases or liquids.
Recently, some progress toward the modelling of
non-Boussinesq density currents of arbitrary density ratios
and initial depth ratios has been made. Ungarish (2007)
revisited the one-layer shallow-water (SW) model (which is
the backbone of the abovementioned models for Boussinesq
flows) for the prediction of the shape and propagation of
high-Reynolds number density currents. The model applies
for both the constant-speed (slumping) and self-similar
regimes over the complete range of density ratio and initial
depth ratio. The model provides a useful tool for
understanding the dynamics of density currents, since
Boussinesq and non-Boussinesq currents are treated in a
unified manner.
The realization of non-Boussinesq density currents
in the laboratory, for a wide range of parameters, is a
difficult and expensive task, because exotic materials and
appropriate containers are necessary for density ratios not
close to 1. This is the reason why very few setups were ever
activated and for full-depth lock release mostly. The
numerical simulations are, presently, the only effective
means for gaining novel pllic 'ii,'ncl ', gk .il" knowledge on
the non-Boussinesq flow field, and for testing
systematically the available theoretical models.
We consider the propagation of a density current of
density Pc and initial depth ho into an ambient fluid of
density Pa and initial depth H (figure 1). Depending on the
sign of Pa-Pc, we refer to a bottom (heavy) density current
(Pa-pc<0) or a top (light) density current (Pa-pc>0). The
current is released from a lock of initial position xo and
height ho. In this paper, we report results of a series of
two-dimensional simulations of planar density currents for a
wide range of density ratios 10

depth ratios in the range 1< H* < 50 (where H* = H / h),
and lock aspect ratios 0.5 < A < 18.75 (with = i,. li,,'


Nomenclature


gravitational constant (m s-2)
density (kg m 3)
height of the ambient (m)
height of the current (m)
velocity (m s-1)
characteristic slumping time
characteristic time of head formation


Greek letters

p density (kg m 3)
v kinematic viscosity of the heavy fluid (m2 s-1)

Subsripts


current
ambient
initial (t=0)
nose


Ax/h,= 1/200 (1/64)




Ambient p,

----
I Current Pc
I ': ---- ^\ __


0 eX


H
I 71 with

1 5h,
A/ho = 1/160
i 0 (1/160)
I A!, 1/4o80
0 (1/160)


L/h 12 5
(375)


Figure 1: Physical configuration used in the present work
and spatial resolution of the Navier-Stokes simulations. The
dashed line represents the initial separation between fluids,
while the solid line represents the interface at later times.
Here, the gravity vector is defined as
g= g(pa- Pc)/lp, Ple thus both bottom and top
density currents propagate on z=0. The spatial resolution is
identical for all A except 2=18.75 for which the resolution is
given in parenthesis.


Shallow-water model

In the following, we briefly describe the assumptions and
equations used in the shallow-water model used in the
present work to obtain, among others, the height hN and the
velocity UN of the current's nose during the constant-speed
initial phase. We refer the reader to Ungarish (2007) for
more details about the model.
In the present model, we consider incompressible,
immiscible fluids, and assume that the viscous effects are
negligible (in both the interior and the boundaries). The
thickness of the current is h(x,t) and its horizontal velocity
(z-averaged) is u(x,t). Initially, at t=0, h=ho and u=0. We
assume a shallow current which, formally, implies a
non-small, but not clearly specified, value of A. To close the
system of equations an additional boundary condition is
specified at the front of the current located at XN. The model
makes use of the condition


UN=(ghNI Pc a I/pa)1Fr,


where the Froude number, Fr, is a function of the depth
ratio h/JH and is taken to be the Benjamin's (1968) relation
given by
S1/2(2)
Fr(hN [ (2- hN/H)(1- hN/H) (2)
FL (1+ h,/H)

Further assuming that the instantaneous height of the
current is constant along its body and that the energy in the
domain cannot increase, one can match the shallow-water
solution of the forward-propagating characteristic with
Benjamin's front condition to obtain the height and velocity
of the fronts. The height of the current is then given by the
following implicit equation


h Fr(hNIH)
L ho


2 L1- 1
P, ho0


Paper No





Paper No


where the Froude number relation is given in (2). If we
apply the additional energy constraint, the maximum height
of the current can be limited to H/2 and we obtain
hN = min(h,, H / 2) The velocity of the current can then be
determined from (1).


Navier-Stokes solver

The numerical approach used here is the JADIM code
developed at IMFT, Toulouse. Briefly, this code is a
finite-volume method solving the three-dimensional,
time-dependent Navier-Stokes equations for a
variable-density incompressible flow (of arbitrary density
variations), together with the density equation, assuming
molecular diffusivity to be negligibly small. The transport
equation of the density is solved using a modified Zalesak
scheme (mixed low-order/high-order scheme, Zalesak
1979). Momentum equations are solved on a staggered grid
using second-order centred differences for the spatial
discretization and a third-order Runge-Kutta /
Crank-Nicolson method for the temporal discretization. The
incompressibility condition is satisfied using a
variable-density projection technique. The overall
algorithm is second-order accurate in space and first-order
accurate in time. We refer the reader to Bonometti et al.
(2008) and Hallez & Magnaudet (2009) for more details on
the equations solved and the numerical technique.
In this approach, the transport equation of the
density is hyperbolic. This is equivalent to choosing an
infinite Schmidt number, defined as the ratio of the
kinematic viscosity to the molecular diffusivity. Although
no physical diffusivity is introduced, the numerical
thickness of the interface is not strictly zero as it is typically
resolved over three grid cells (Bonometti & Magnaudet
2007). Therefore a finite effective Schmidt number can be
estimated, which depends somewhat on the Reynolds
number and on the degree of spatial resolution. Based on
extensive tests of measurement of the interface thickness,
Bonometti & Balachandar (2008) estimated that, for similar
Reynolds number and spatial resolution as those used in the
present work, the effective Schmidt number was of O(103).
Thus, the numerical approach allows for the processes
mixing and entrainment as one could observe in the
transition region between weakly diffusive fluids
(salted/fresh waters for instance). Note however, that our
simulations didn't detect any significant entrainment in the
range of density ratios investigated. We note in passing that
we recorded the temporal evolution of the overall
mechanical energy. In all the cases, the relative variation of
the total energy remains negligibly small during the entire
duration of the simulation (less than 0.1%), indicating that
the effect of numerical diffusion is marginal.


Numerical setup

The simulations reported here are two-dimensional and are
performed within a rectangular (x,z) domain LxH large. In
the following we set L=12.5ho for all but the largest lock
aspect ratio considered here; namely when 2=18.75 we
choose L=37.5ho (this configuration will be referred to as
the long domain case as opposed to the short domain case,


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

and the characteristics of the long domain grid are given in
parenthesis). We have paid careful attention to spatial
resolution in order to ensure grid-independent results. In
particular, the grid is refined near the bottom boundary so as
to accurately capture the front of the current, which is
highly elongated in the high-density ratio configurations.
We use a 2500x300 (2400x300) uniform grid with a spacing
of AJ /,,=1/200 (1/64) in the x-direction. In the z-direction,
the domain is divided into three regions. In the region
0< z<0.1ho, a uniform spacing of A- ii,,=1 4,,I (1/160) is
used, while a spacing of J- 1i,,= 1/160 (1/160) is used over
the region 0.1ho current and a significant portion of the ambient fluid
entrained by the current). Finally, larger cells are used above
z=1.5ho, following an arithmetic progression. Free-slip
boundary conditions for the velocity (unless otherwise
specified) and zero normal gradient for the density are
imposed on the top, bottom and lateral boundaries.
The computations to be described below were run
at a prescribed Reynolds number Re = Uho / v of 2.5x104,
where U = g'h and g'=glpc-pal/max(pc, Pa) is the
reduced gravity. We further assume the dynamical viscosity
to be the same for both fluids (see e.g. appendix B of
Bonometti et al. (2008) for a discussion of this assumption).
Here we varied the density ratio in the range
10 4 < Pc / p <104 Note that in order to keep the Reynolds
number constant while the density ratio varies, we modify
the viscosity of the fluids accordingly.


ioi-1o -o


0
2.5


F, +10


ohm


I I

I J


I I I I


Figure 2: Temporal evolution of the shape of a top density
current shadowgraphh) for H*=50 and pc/Pa=10-3. The time
interval between successive views is AtJg/ h = 2.827
Axes are scaled by ho.





Paper No


Large initial lock aspect ratios

In this section the initial lock ratio is fixed at the largest
value A = 18.75. The shape of Boussinesq density currents
of arbitrary initial depth ratios has been extensively
described in the literature as well as that of non-Boussinesq
currents in the lock-exchange configuration. However, the
description of non-Boussinesq top density currents in a deep
ambient is much less documented. Therefore, we plot in
figure 2 the temporal evolution of the shape of a top density
current (H*=50 and pc/Pa=10-3). Soon after the removal of
the gate, the head of the current is clearly visible and takes a
rounded shape. At the same time, a surface perturbation is
formed behind the head. The distance separating the surface
perturbation from the current head is observed to grow in
time while the height of the current remains approximately
constant in between. Therefore, for the times considered
here, the shape of the top density current can be
decomposed into a head, a surface perturbation (note that
some small vortical structures are discernable at the
junction with the head) and a tail further downstream.
Since t>>1 here, it is reasonable to compare the
speed of propagation obtained from the Navier-Stokes
simulations with that predicted by the one-layer
shallow-water theory in the constant-speed phase. The front
velocity obtained from the Navier-Stokes simulations
(symbols) is compared in figure 3 with the shallow-water
model (lines), for density currents of various density ratios
and two different initial depth ratios, namely H*=I and 50,
respectively. Note that intermediate initial depth ratios were
also investigated and the results were in essential agreement
and therefore, for clarity, they are not presented here. The
front velocity was computed as follows. For each simulation,
we record the temporal evolution of the front position XN
defined as the maximum value of x for which the equivalent
height h is non-zero, where h is defined as (Marino et al.
2005; Cantero et al. 2007)


h(x,t)= (p(x,z)-p )/(P,-P,)dz


Here p is the local value of the density field. The
instantaneous speed of propagation UN is then computed as
UN=dxN/dt. For all the 2=18.75 cases, an acceleration phase
followed by a nearly constant phase (slumping) is observed.
The agreement is fairly good in the case H'*=
(lock-exchange configuration) for the complete range of
density ratio shown here. Indeed, the front velocity of top
density currents ( 104 < c / p < 1 ) predicted by the
shallow-water model agrees within 1% with the simulations
while for bottom density currents (1< P / Pa < 104 ) the
maximum discrepancy is less than 10%. Such discrepancy
was previously observed in Bonometti et al. (2008) for
101 < Pc / Pa < 6 x101, and was attributed to dissipation
stemming from vortical structures generated near the front
of the current.
For H*=50, good agreement is observed for bottom
density currents ( 1p, / p, <104 ). In line with the
shallow-water model prediction, the results become
independent of H* as the density ratio increases. For
instance, the front velocities of the H*=I and


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

H*=50-currents are nearly identical for Pc / pa = 103. The
situation is different for top density currents. As shown in
figure 3, we observe an increasing departure between the
predicted (solid line) and computed velocities as the density
ratio is decreased. The shallow-water model overestimates
the front velocity by at most 25% for the smallest density
ratios investigated here. We also plot for comparison the
experimental results of Rottman & Simpson (1983) in the
Boussinesq limit at intermediate initial depth ratios, namely
H*=4 and H* 15 (see the diamond and circle symbols). A
monotonic increase in the velocity is observed from H*= 1 to
H*=50. Comparison between the experimental results and
the present computational results is in general quite good.
The fact that the largest error is observed for small
density ratios could be expected. First, the shallow-water
one-layer model becomes less and less accurate when
H'p / p, decreases because the inertia of the return flow
in the ambient (which is neglected in the shallow-water
model) increases like (H Pc/pa)- More precisely, Ungarish
(2007) estimated that the effects of the ambient fluid on the
dynamics of the density currents can be neglected provided
(H Pc/Pa)- <1/2 approximately. For H*=I this leads to the
condition pa,/P>2. It can therefore be expected that the
agreement between the simulations and the model breaks
down for pc/pa<2. However, an additional energy constraint
arises for p/lpa<1.373 (that is the flow is choked at half the
initial depth). This sets the velocity of the current to be that
imposed by the front condition, and as a result the
agreement is excellent (see the comparison between the
dashed line and the squares in figure 3). For H*=50, the
condition for neglecting the ambient fluid dynamics reduces
to p,/pa>4x10-2. This is in reasonable agreement with our
results for which the discrepancy is maximum for density
ratios in the range p,/pa<4x10-2 approximately. The fact that
the speed discrepancy is at most 25% for a density ratio that
is 400 times smaller than the limit of validity is actually an
indication of the robustness of the shallow-water one-layer
model. This robustness calls for some further theoretical
and numerical investigation. Overall, we may conclude that
one possible reason for the discrepancy between the
shallow-water model and the computed velocities of top
density currents in deep ambient is the fact that the model
neglects the fluid motion in the ambient fluid. Thus, under
such conditions a two-layer model, where both the motion
of the current as well as the ambient is taken into account,
can be expected to perform better.
Second, there were indications that light currents
are more dissipative than heavy currents (Birman et al.
2005; Bonometti et al. 2008). This trend, inferred from
full-depth lock releases, was carefully investigated in the
present systems. The Navier-Stokes results indicated that
dissipative effects are larger for top currents than for
Boussinesq or bottom currents. Indeed, for pc/Pa=102
(H*=10), the dissipation energy was found to be of 10% that
of the kinetic energy; this is 4-5 times greater than for
Pc/Pa=1.01 and ppa=102. This suggests that for longer times,
viscous effects may influence the propagation of the top
current, inasmuch that the assumption of negligible viscous
effects is not valid anymore.





Paper No


.UN



0.5 ---------



104 10 102 10 100 101 102 103 104


Figure 3: Slumping front velocity as a function of the
density ratio. The solid (resp. dashed) line is the prediction
of the shallow-water model for H*=50 (resp. H*=1).
Triangles (resp. solid squares) are Navier-Stokes results for
H*=50 (resp. H*=1). Crosses are numerical results of
Birman et al. (2005) (H*=I). Experimental results of Lowe
et al. (2005) (open square, H*=I) and Rottman & Simpson
(1983) (diamond, H*=4; circle, H* 15) are also plotted for
comparison.



Third, it has been shown that the shallow-water prediction is
quite sensitive to the choice of model for the front condition
(2) in the case of high H* and small plPa (Bonometti &
Balachandar 2010). For instance for Pc/Pa=102, if one
chooses the well-known empirical front condition proposed
by Huppert & Simpson (1980) instead of (2), the value of
front velocity predicted by the shallow-water model agrees
within 2% (instead of 25%) with the Navier-Stokes results.
It should be stressed that it is therefore difficult to clearly
disentangle the contribution of the fluid motion in the
ambient (including inertia and dissipation) to that of the
model for the front condition used in the shallow-water
model on the observed discrepancy, so that no definite
conclusion can be drawn at the present time. One alternative
is to solve the more general shallow-water two-layer
equations for arbitrary density ratios and initial depth ratios,
so the inertia in the ambient fluid is included. The extension
of the one-layer arbitrary density ratio shallow-water
model to two layers is not straightforward since it raises
some non-trivial numerical issues. Overcoming these issues
requires a substantial effort which is beyond the scope of the
present work.

Arbitrary initial lock aspect ratios

The evolution of density currents is now considered for
different values of the lock aspect ratio A= ,, /I,, ranging
from 0.5 to 18.75. The density ratios investigated here range
from top density currents to bottom density currents, while
the initial depth ratio is set to H'= 10. From the inspection of
the temporal evolution of the front velocity, three regimes
were observed (not shown here). (i) For large aspect ratio,
greater than some critical value, say A2t, the propagation of
the current was observed to be independent of the lock


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

aspect ratio. For example, for p,/pa>1, the 2=18.75 and
2=6.25 curves of uN(t) collapse. (ii) There exists an
intermediate regime where the front velocity vs. time
depends on the lock aspect ratio, but still a near constant
velocity can be observed. (iii) For small aspect ratio, there is
no constant velocity. This A-dependence is illustrated in
figure 4, where the temporal evolution of the front speed is
displayed for A ranging from 0.5 to 18.75 (,/Pa=10-2). For
comparison, we plotted as solid line the prediction of the
shallow-water model (1)-(3) in the slumping regime.
Clearly, the discrepancy between the Navier-Stokes results
and the shallow-water prediction increases as A is decreased.
These observations indicate that the speed of propagation of
density currents is sensitive to the value of A. We may thus
conclude that the front velocity is not only a function of Re,
Pc/Pa, and H* but of A. It essentially decreases when A is
decreased and the discrepancy is more significant for top
density currents than for bottom density currents.
In the limit of light-top density currents, we derive a
simple model showing that the front velocity UN scales like
e24. As mentioned above, the shape of the top density
currents is characterized by a head of rounded shape
followed by a nearly flat body. The angle made by the front
of the current and the horizontal boundary is close to the
value 7/3, as predicted by Benjamin (1968). Therefore, the
volume 6 (per unit width) occupied by the fluid in the head
of the current can be estimated as O= (4r- 3) h where
hN is the maximum height of the head. Here, we further
assume that if the lock aspect ratio 2 is smaller than a
critical value, say Ac2t, then the fluid in the current is
entirely located inside the head of equivalent volume and
angle 7/3 (with respect to the horizontal boundary). Let
V=xoho be the initial volume (per unit width) of the current.
We can write V = 6 which yields the following expression
for the dimensionless height of the current


hN/ho (4 J-%ll 2 A


We argue that (1) is still a valid approximation for the speed
of propagation, and hence, upon substitution of (6), we
obtain


UN (= Frf I (1 4LrZ).
U


The solution (7) is plotted in figure 5 (dash-dot line)
and compared with the front speed obtained from the
Navier-Stokes simulations (instantaneous values taken in a
prescribed time interval). Also plotted is the slumping
velocity predicted by the shallow-water theory (1)-(3).
Predictions of (7) are in reasonable agreement with the
computational results at small lock aspect ratios. As
expected, as A increases the front velocity becomes
independent of the lock aspect ratio and the speed of
propagation is close to that predicted by the shallow-water
theory. Note also that in the present configuration
(/Pa=10-2), the critical value of A for which the front
velocity given in (7) is equal to that predicted by the
shallow-water theory is Ac, (4-,f )= 11.






Paper No


UN
gh_ 0.6
u I0. 6 . . .. .. .. .. ..
N- -'- -.-- -- '--- -- --

0.4

0.2

0 5 10 15 20
t

Figure 4: Temporal evolution of the front velocity uN for
various lock aspect ratios A (pPa=10 2): 2=18.75;
........, 2=6.25; -------- =1; -.-.-.--, 2=0.5. The
horizontal lines with circle represent the slumping velocity
predicted by the shallow-water theory (eq. 1-3).


UN
-gb0


Figure 5: Ranges of instantaneous front velocity (taken in a
time range Tj simulations vs. 1/4 for top density currents (p/lpa=10-2;
H =10). The solid vertical solid bars are UN in the time range
[TI/T=2, TF/T=15]. For comparison we plotted as vertical
dotted bars UN in the time range [TIIT=0.6, TFIT=15] which
includes the end of the acceleration phase. Observe that the
trend is similar despite the amplitude of the variations is
different. solution (7); ----, predicted slumping front
velocity from the shallow-water theory (1)-(3). Note that in
(7), we used Fr=1.27 as obtained from (2)-(3).


Summary


We carried out a numerical investigation of high-Reynolds
number constant volume non-Boussinesq density currents
propagating over a horizontal flat boundary in a deep
ambient. The goal is to shed light on the influence of the
density ratio and lock aspect ratio on the shape and
dynamics of the current. Solutions of the shallow-water


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

equations were also used for comparison with the
Navier-Stokes simulations.
In the accepted description, the high Reynolds
number density current created by lock-release, depends on:
(1) In the Boussinesq case, one free parameter, the initial
depth ratio H*; and (2) In the non-Boussinesq case, on two
parameters H* and the density ratio, plpa. Our results
highlight an additional parameter: 2= .,, i/,,. which may play
a role in both Boussinesq and non-Boussinesq cases. It was
observed that the shape and speed of propagation of density
currents are influenced by the lock aspect ratio 2, if A is
below a critical value A,,. The critical value of A depends
primarily on the density ratio plpa, and to a lesser extent, on
the initial depth ratio H*. In the specific case of light-top
density currents, we developed a simple model predicting
the dependence of the front velocity on A (for A speed of propagation was found to evolve as 14, in
reasonable agreement with the Navier-Stokes results.
This work is just a preliminary report, and a more
sound and extended investigation will be presented in a
journal paper which is now in preparation.


Acknowledgment

MU thanks the University of Florida at Gainesville for
hosting his academic visit during which a part of this
research was carried out.


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