theory and experiments - cnr · the earliest recorded experiment of a gravity current by marsigli...

Gravity-driven flowsTheory and Experiments

Paul Linden

[email protected]

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


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


Introduction

history

natural and laboratory gravity currents

reduced gravity

driving forces

frontogensis


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


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


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


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


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”


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.


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


FrontogenesisUniform density gradient

Isopycnals remain straight as they tilt towards the horizontal. The flow is stable


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)


Frontogenesis

Density increases from clear to blue to yellow to red


Dimensional analysis

non-dimensional parameters

Reynolds number

constant-volume release - 2D

scaling analysis with Froude number


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


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.


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


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)


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


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


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)


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


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


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.


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


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)


Full-depth lock release

Comparison with Benjamin’s potential flow solution


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


FINE


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

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