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Gravity-driven flowsTheory and Experiments
Paul Linden
Department of Mechanical & Aerospace Engineering
UC San Diego
Buoyancy-driven flows – p.1/38
OutlineLecture I – Gravity currents in uniform environments
Introduction – natural and laboratory gravity currents,
motion driven by density gradients, frontogenesis
Dimensional analysis: constant-speed and similarity phases
Froude numbers - theories of Yih, von Kármán and
Benjamin
Comparison with experiment
Lecture II – Gravity currents in uniform environments
Energy-conserving theories
Shallow water theory
non-Boussinesq currentsBuoyancy-driven flows – p.2/38
OutlineLecture III – Gravity currents in stratified environments
Gravity currents in stratified environments
Intrusions in a two-layer fluid
Intrusions in a constant N fluid
Stratified intrusions
Buoyancy-driven flows – p.3/38
Gravity CurrentsLecture I – uniform environments
Outline
Introduction – natural and laboratory gravity currents,motion driven by density gradients, frontogenesis,
Dimensional analysis: constant-speed and similarityphases
Froude numbers - theories of Yih, von Kármán andBenjamin
Comparison with experiment
Buoyancy-driven flows – p.4/38
Introduction
history
natural and laboratory gravity currents
reduced gravity
driving forces
frontogensis
Buoyancy-driven flows – p.5/38
The first experiment – 1681
The earliest recorded experiment of a gravity current by Marsigli (1681). Salt water was
placed on the right side of the barrier and fresh water on the left. When openings were
made at the top and bottom, two gravity currents were formed, with a fresh current at
the top and a salty current at the bottom. Marsigli used this experiment to demonstrate
the exchange flow through the Bosphorus
Buoyancy-driven flows – p.6/38
Natural gravity currents
Ash-laden gravity current from the erup-
tion of Mount Pinatubo in 1991. This
amazing photograph was taken by Al-
berto Garcia and is reproduced by per-
mission of the National Geographic. The
occupants in the vehicle survived
A pyroclastic flow resulting from the
eruption of Mount Unzen, Japan in
1990. The volcano had been inactive for
almost 200 years before an active period
from 1990-1995
Buoyancy-driven flows – p.7/38
Economically important gravity currents
Record of the sea-breeze at La Jolla.
This temperate wind keeps the coastal
zone cool and property prices high
A spill of LNG on the sea surface – the
cloud is visible as a result of condensa-
tion of water vapour
Buoyancy-driven flows – p.8/38
Laboratory gravity current
The pressure is greater under the dense blue fluid – caused by salt dissolved in water –
providing a horizontal force from left to rightBuoyancy-driven flows – p.9/38
mov00251.mpgMedia File (video/mpeg)
Lab versus Nature
A dust storm created by cold air flowing
out from under a thunderstorm. This
photograph was taken in Leeton, NSW,
Australia
A saline laboratory gravity current flow-
ing into fresh water. The current is
made visible by milk added to the salt
water. The lobes and clefts are clearly
visible. The three dimensional structure
persists behind the front and affects the
structures at the top of the current
Buoyancy-driven flows – p.10/38
Lobes and clefts
Lobes and clefts caused by gravitational instability at the front of the current Simpson
(1972)Buoyancy-driven flows – p.11/38
mov00260.mpgMedia File (video/mpeg)
Reduced gravity
!"#
!$#
A schematic of a lock exchange exper-
iment. In this case fluid of density ρUis separated by a vertical partition – the
lock gate – from denser fluid with den-
sity ρL. Both fluids are initially at rest.
When the gate is removed a dense grav-
ity current will flow along the bottom to
the right and a buoyant current will flow
along the top to the left
∂p
∂z= −gρ
Integrate down from the surface
p = −gZ H
0
ρ dz
∆p = g(ρL − ρU ) = g∆ρH
Now
pressure difference = mass x acceleration/area
Since mass = density x volume,
∆p = ρHa
where a is the acceleration. Hence
a = g∆ρ
ρ≡ g′
Reduced gravity - “g prime”
Buoyancy-driven flows – p.12/38
Driving forces
Compositional gravity currents
Density difference produced by
dissolved solute - e.g. salt in the sea
temperature - in a gas ∆ρρ
= ∆TT
Particle-driven gravity currents
Density difference produced by suspension of particles∆ρρ
=ρp−ρf
ρfφ
φ = volume concentration of particles
Boussinesq fluid
∆ρρ
Frontogenesis
The motion of isopycnal surfaces under
the action of gravity for a fluid with a
constant horizontal density gradient. (a)
The initial condition with vertical isopy-
cnal surfaces. The arrow indicates the
generation of baroclinic vorticity. (b)
The position of the isopycnal surfaces
at a later time. The isopycnals remain
straight as a result of the constant ver-
tical shear.
The motion of isopycnal surfaces under
the action of gravity for a fluid with a
constant horizontal density gradient in
an experiment. The isopyncals remain
straight and all rotate at the same rate
as the flow evolves. There is no evidence
of any instabilities in the flow.
Buoyancy-driven flows – p.14/38
FrontogenesisConsider density ρ = ρ(x) only. 2D flow
u = (u, 0, w)∂ρ
∂t+ u
∂ρ
∂x= 0
∂u
∂x+
∂w
∂z= 0
„
∂
∂t+ u
∂
∂x
«
∂ρ
∂x=
∂w
∂z
∂ρ
∂x
If ∂ρ∂x
= ∂ρ∂x
|0 constant, then w = 0 and∂u
∂t= − 1
ρ0
∂p
∂x
∂p
∂z= −gρ
Cross differentiate
∂2u
∂t∂z=
g
ρ0
∂ρ
∂x|0
Since continuity implies that u = u(z),
this equation may be integrated to give
u =g
ρ0
∂ρ
∂x|0zt
where the flow has been assumed to
start from rest and that u(0) = 0
When the horizontal density gradient is
constant in space
∂ρ
∂z= − 1
2
g
ρ0
„
∂ρ
∂x|0
«2
t2
The gradient Richardson number
Ri ≡− g
ρ0
∂ρ∂z
“
∂u∂z
”2=
1
2
Buoyancy-driven flows – p.15/38
FrontogenesisUniform density gradient
Isopycnals remain straight as they tilt towards the horizontal. The flow is stable
Buoyancy-driven flows – p.16/38
FrontogenesisNon-uniform density gradient
(a) The initial conditions with vertical
isopycnal surfaces. (b) The position of
the isopycnal surfaces at a later time.
The larger vorticity generation on the
left causes the isopycnals between the
two regions of constant density to con-
verge producing a front on the lower
boundary.
Sequences from a laboratory experiment
showing frontogenesis. L & Simpson
(1989)
Buoyancy-driven flows – p.17/38
Frontogenesis
Density increases from clear to blue to yellow to red
Buoyancy-driven flows – p.18/38
Dimensional analysis
non-dimensional parameters
Reynolds number
constant-volume release - 2D
scaling analysis with Froude number
Buoyancy-driven flows – p.19/38
Non-dimensional parametersThe most important non-dimensional parameter for a Boussinesq gravity current is the
Froude number FH defined as the ratio of the current speed U to the long wave speed√g′H
FH =U√g′H
The choice of the length scale H is an important aspect
The second important parameter is the Reynolds number Re
Re ≡ UHν
The effects of diffusion of density are measured by the Peclet number Pe
Re ≡ UHκ
And for non-Boussinesq currents the density ratio
γ ≡ ρUρL
Buoyancy-driven flows – p.20/38
Reynolds number
(a)
(b)
(c)
Shadowgraphs of a dense saline gravity current at different values of the Reynolds
number. In (a) Re ≈ 1000, in (b) Re ≈ 8000, and in (c) Re ≈ 20000.
Buoyancy-driven flows – p.21/38
Dimensional analysisConstant volume release – 2D
x
z g
!U
!L
L0
D
L(t)
H
g´(t) h(t)
Initially
U = F (g′0D)1/2f(t/Ta)
where F is a dimensionless constant
When t >> Ta =p
D/g′0
U = FD(g′
0D)
12
– constant-velocity phase
Current length
L(t) = L0 + FD(g′
0D)
12 t
where FD ≡ U√g′D
is the Froude num-
ber based on the original release height
Later times
U = FD(g′
0D)1/2f(t/Ta, t/TV ).
When t >> TV =L0√g′0D
the total
buoyancy B0 = g′0DL0 (= constant)
becomes important
Dimensions [B0] = L3 T−2
U =2
3cB
130
t−13
L = cB130
t23
Current decelerates as t−1/3 – similarity
phaseBuoyancy-driven flows – p.22/38
Scaling analysisFront Froude number
Conservation of mass
g′(t)L(t)h(t) = cBg′
0A0 = cBB0
Assume front travels with constant local
Froude number
Fh ≡U√g′h
U =dL
dt= Fh(g
′(t)h(t))1/2 = Fh
r
cBB0
L
L(t) = [3
2Fh(cBB0)
1/2t + (L0)3/2]2/3
L(t)
L0= [
3
2FhcB
12 t/TV + 1]
2/3
t > TV
L(t)
L0≃
„
3Fh
2
«
23
cB13 (t/TV )
2/3
similarity phase
Buoyancy-driven flows – p.23/38
The viscous phaseViscous time scale
Tν =νLν
2
g′νhν3
U = F (g′νhν)1/2f(t/Ta, t/TV , t/Tν)
When t >> Tν balance between viscous forces and pressure gradient ν∇2u ∼ ∇p/ρ
ν
h2dL
dt=
cνg′νh(t)
L(t)
L(t)h(t) = cAAν
L(t) = [5cνcA
3g′νAν3
νt + Lν
5]1/5
Current decelerates as t−4/5
For a complete theory of viscous currents (honey on toast or lava flows see Huppert
1982)
Buoyancy-driven flows – p.24/38
Transitions between the phasesLaboratory experiments
Shadowgraph and data showing the transitions between the constant-velocity, similarity
and viscous phasesBuoyancy-driven flows – p.25/38
Froude number
Theories of
Yih 1938
von Kármán 1940
Benjamin 1968
Buoyancy-driven flows – p.26/38
Froude numberYih 1938
!"#
!$#
A sketch of the idealised of a Boussinesq
lock release with symmetrical light and
heavy currents.
Boussinesq current (i.e. ρL ≈ ρU ),then symmetry implies that the current
will initially occupy one-half the depth.
In a time ∆t
PE gained by the lighter fluid =1
4gρUH
2U∆t
PE lost by denser fluid =1
4gρLH
2U∆t.
KE gain = 14(ρU +ρL)HU
2U∆t.
U =
s
g(ρL − ρU )H2(ρU + ρL)
.
Boussinesq case ρU ≈ ρL
FH ≡U√g′H
=1
2Buoyancy-driven flows – p.27/38
Froude numbervon Kármán 1940
A particle–driven gravity current
U
!L
!U
h
g
O C B
A
An idealized model of a perfect fluid
gravity current.
Ideal flow - Bernoulli’s theorem
p +1
2ρq2 + gρz = constant
pO = pA + gρUh + 12ρUU2
Assume no flow inside the current
pO = pA + gρLh
U2 = 2gρL − ρU
ρUh = 2g
1 − γγ
h
Fh =U
p
g(1 − γ)h=
s
2
γ
Boussinesq current (γ ∼ 1)
Fh =U√g′h
=√
2
Compare with Yih’s result H = 2h ⇒
Fh =U√g′h
=U√g′H
r
H
h=
√2
2
Buoyancy-driven flows – p.28/38
Froude numberBenjamin 1968
U
h
uU
H
!L
!U
B O C
D E
Control volume moving with current
UH = uU (H − h)
Along BE
p =
8
<
:
pB − gρLz, 0 < z < h,pB − gρLh − gρU (z − h), h < z < H
Along CDp = pC − gρUz
Conservation of horizontal component of the momentum fluxE
Z
B
pdz +
EZ
B
ρu2dz =
DZ
C
pdz +
DZ
C
ρU2dz
u =
8
<
:
0, 0 < z < h,
uU , h < z < H
FH ≡U√g′H
=p
f(h) f(h) =h(2H − h)(H − h)
H2(H + h)
Buoyancy-driven flows – p.29/38
Benjamin’s theoryInfinite depth
In the limit H → ∞
U2
g′h= Hf(h) =
h(2H − h)(H − h)H(H + h)
→ 2
Hence the Froude number based on the current height
Fh =√
2
in agreement with von Kármán
Buoyancy-driven flows – p.30/38
Benjamin’s theoryFroude number
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
The Froude number FH and the dimensionless volume fluxQ√
g′H3plotted against the
dimensionless current depth hH
Buoyancy-driven flows – p.31/38
Benjamin’s theoryEnergy-conserving current
U
h
uU
H
!L
!U
B O C
D E
Along the upper boundary ED
pE +1
2ρU uU
2 = pD +1
2ρUU
2
pE −pD = pB −pC −g(ρL−ρU )h.
pB − pC = −1
2ρUU
2
1
2ρUuU
2 = g(ρL − ρU ).
Continuity ⇒
U2 = 2g′h(H − h)2
H2.
Two solutionsh
H= 0 or
h
H=
1
2.
Energy-conserving current occupies one-half the depth
F ≡ U√g′H
=1
2.
Buoyancy-driven flows – p.32/38
Benjamin’s theoryProperties of the energy-conserving current
!"#$%&'()"*#$+,"%'-.'
/01'*2##3'-.'
h
H=
1
2
F ≡ U√g′H
=1
2.
Froude number based on current
height
Fh =U√g′h
=1√2⇒ subcritical lower layer
FU ≡uU
p
g′(H − h)=
√2 ⇒ supercritical upper layer
Two-layer flow with FL = Fh implies
FU2 + FL
2 = 1
Maximum speed occurs at depth hm = 0.347HBuoyancy-driven flows – p.33/38
Comparison with experiments
half-height currents
comparison with Benjamin’s shape
full-depth lock releases
Froude numbers
Buoyancy-driven flows – p.34/38
Half-height currents
Air cavity in
a rectangular
horizontal
duct: Gard-
ner & Crow
(1970)
Red line shows effective depth. Blue lines give h/H = 0.5 and h/H = 0.347.
The effective
depth h:
Shin et al.
(2004)
Buoyancy-driven flows – p.35/38
Full-depth lock release
Comparison with Benjamin’s potential flow solution
Buoyancy-driven flows – p.36/38
Full-depth lock release
t*= 0.0
t* = 0.4
t*= 1.2
t*= 2.3
t*= 3.9
t*= 4.7
t*= 5.9
t*= 7.0
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t*
x / H
t∗ ≡ ts
H
g′
F = 0.48
Buoyancy-driven flows – p.37/38
FINE
Buoyancy-driven flows – p.38/38
OutlineOutlineGravity CurrentsIntroductionThe first experiment -- 1681Natural gravity currentsEconomically important gravity currentsLaboratory gravity currentLab versus NatureLobes and cleftsReduced gravityDriving forcesFrontogenesisFrontogenesisFrontogenesisFrontogenesisFrontogenesisDimensional analysisNon-dimensional parametersReynolds numberDimensional analysisScaling analysisThe viscous phaseTransitions between the phasesFroude numberFroude numberFroude numberFroude numberBenjamin's theoryBenjamin's theoryBenjamin's theoryBenjamin's theoryComparison with experimentsHalf-height currentsFull-depth lock releaseFull-depth lock release