Morris R. Flynn, Dept. of Mech. Eng., Univ. of Alberta

Gravity current flow in two-layer stratified media

Primary funding:

CAIMS 2016, Geophysical Fluid Dynamics mini-symposium

Collaborators/STUDENTS: Brian A. Fleck, ALEXIS K. KAMINSKI, Paul F. Linden, MITCH NICHOLSON, David S. Nobes, RYAN M. SAHURI, Bruce R. Sutherland, ALAN W. TAN, Marius Ungarish

Gravity currents in the atmosphere• Gravity currents are primarily horizontal flows driven by density differences

and are ubiquitous features of many industrial and environmental processes

• Atmospheric manifestations include dust storms (“haboobs”), thunderstorm outflows and “microbursts”; all represent a significant threat to aircraft (Linden & Simpson, 1985)

Phoenix sandstorm (July 5, 2011)

Photo credits:http://dailyshot.homestead.comhttp://www.stormeyes.org/tornado/SkyPixhaboob.htmhttp://www.damtp.cam.ac.uk/user/fdl/people/jes14

Wet microburst

Thunderstorm outflow

Gravity currents in marine environments

• Marine manifestations include river plumes (Nash & Moum 2005), which influence coastal ecology, pollution transport, etc.

Photo credits: http://www.glerl.noaa.govhttp://fvcom.smast.umassd.edu

Grand River plume (Grand Haven, MI)

Changjiang River plume (East China Sea)

Introduction

Q? If gravity currents are driven by density differences, how can we relate the gravity current front speeds to these differences?

Q? Does the front speed also depend on geometric parameters such as the gravity current height?

• High-Re flow (ignore viscous dissipation)

• Rectilinear geometry, channel of finite height (channel of infinite height is a straightforward extension)

• Non-rotating reference frame

• Density difference due to difference of composition, temperature or dissolved salt, not (sedimenting) particles

Front speed is then constant at least for early times

A flow this ubiquitous merits detailed attention

Will address these questions using the following assumptions:

• The flow of a high-Re gravity current was examined theoretically by T. Brooke Benjamin in 1968* who applied a Galilean change of reference frame (gravity current front stationary, ambient in motion)

Benjamin’s 1968 theory

Mass continuity (steady state):

UH = u(H � h)

) u =UH

H � h

One eqn. but three unknowns:O A

D

B

C

u

U

h

ρ

ρ1

0 Hz

x

*T.B. Benjamin, J. Fluid Mech., 31 (1968) -- Cited > 600 times

U, u, h

• No external forces acting on the flow, therefore “flow force” is conserved, i.e.

Benjamin’s 1968 theory

O A

D

B

C

u

U

h

ρ

ρ1

0 Hz

x

• Apply this result far up- and downstream where mixing is negligible and pressure, p, is hydrostatic

Z D

A(p+ �v2) dz =

Z C

B(p+ �v2) dz

Z(p+ �v2) dz = constant

v - horizontal velocity

Benjamin’s 1968 theory

O A

D

B

C

u

U

h

ρ

ρ1

0 Hz

xZ D

A(p+ �v2) dz =

Z C

B(p+ �v2) dz yields

Fr2 =U2

g0H=

h(H � h)(2H � h)

H2(H + h)

0 0.2 0.4 0.60

0.2

0.4

h/H

Fr

Fr Froude number (non-dim. front speed)-

g0 = g

✓�0 � �1

�1

◆- reduced gravity

Benjamin’s 1968 theory

O A

D

B

C

u

U

h

ρ

ρ1

0 Hz

x

0 0.2 0.4 0.60

0.2

0.4

h/H

Fr

• Solutions with h/H > 0.5 have negative dissipation and are therefore unphysical

• Realized value of h/H depends on initial condition

• If gravity current fluid initially spans the entire channel depth, h/H = 0.5

Density-stratified ambient• Benjamin’s 1968 theory assumes a uniform ambient, but this is inaccurate

in most environmental and many industrial contexts

• Density stratification introduces a myriad of new complications: dynamic coupling may arise between the gravity current front and internal or interfacial waves

• When the ambient is stratified, gravity current may still propagate along lower (or upper) boundary, or it may propagate as an intrusion inside the stratified fluid

• Benjamin’s 1968 theory assumes a uniform ambient, but this is inaccurate in most environmental and many industrial contexts

• Density stratification introduces a myriad of new complications: dynamic coupling may arise between the gravity current front and internal or interfacial waves

• When the ambient is stratified, gravity current may still propagate along lower (or upper) boundary, or it may propagate as an intrusion inside the stratified fluid

Intrusion flow along a sharp density interface

Density-stratified ambient

Flynn & Sutherland, J. Fluid Mech., 514 (2004)Sutherland, Kyba & Flynn, J. Fluid Mech., 514 (2004)Flynn & Linden, J. Fluid Mech., 568 (2006)

• Benjamin’s 1968 theory assumes a uniform ambient, but this is inaccurate in most environmental and many industrial contexts

• Density stratification introduces a myriad of new complications: dynamic coupling may arise between the gravity current front and internal or interfacial waves

• When the ambient is stratified, gravity current may still propagate along lower (or upper) boundary, or it may propagate as an intrusion inside the stratified fluid

Density-stratified ambient

Gravity current flow in a two-layer ambient

Boundary gravity current

Discharged effluent

Warm upper layer

Cold lower layer

(a)

Cold lower layer

Internal wave

Discharged effluent(b)

Warm upper layer

Effluent discharge in marine environments

From Flynn, Ungarish & Tan, Phys. Fluids, 24 (2012)

Q? How quickly does effluent travel downstream?

Q? How is this motion influenced by the interfacial wave that may propagate ahead of the gravity current front?

Density-stratified ambientRegarding the flow of a gravity current in the context of pollution dispersion leads to the following questions:

Satisfactorily addressing these (and other) questions requires a judicious combination of theory and experiment (laboratory and numerical)

Theoretical model: apply usual change of reference frame (i.e. gravity current front stationary)

Boundary gravity current

Photo credit:Alan W. Tan

Mass balance:

uihi = Uh′

i i = 1, 2

Geometry:h0 + h1 + h2 = h

′

1 + h′

2

h0 + h1 = h′

1 + η

Flow force balance:

1

2U2H + 1

2g′02H

2 = g′12[

h′

1(H −

1

2h′

1) + 1

2h2

2

]

+ 1

2g′01(h1+h2)

2+U2

(

h′21

h1

+h′2

2

h2

)

One equation in three unknowns:

Boundary gravity current - theory

Q? How do we achieve closure?

Z D

A(p+ �v2) dz =

Z C

B(p+ �v2) dz

U, h1, h2

Bernoulli’s equation (layer 1): Bernoulli’s equation (layer 2):

1

2U2 = g′01

h21

h′21

(H − h1 − h2)1

2U2 =

h22

h′22

[g′02(H − h1 − h2) − g′12(h′

1 − h1)]

Applying both equations leads to unphysical multiplicities, however...

Boundary gravity current - theory

Q? How do we achieve closure?

Holler & Huppert, J. Fluid Mech., 100 (1980); Sutherland, Kyba & Flynn, J. Fluid Mech., 514 (2004)

• We choose to apply Bernoulli’s equation once then relate the amplitude of the interfacial disturbance to the other parameters of the problem

η

η

h′

1

η = 1

2(H − h′

1)

H

Boundary gravity current - theory

When the gravity current fluid initially spans the entire channel depth, parameterization is easy!

• We choose to apply Bernoulli’s equation once then relate the amplitude of the interfacial disturbance to the other parameters of the problem

η

η

h′

1H

Closed symbols: lab expts.Open symbols: numerics

Boundary gravity current - theory

η = 1

2(H − h′

1)

When the gravity current fluid initially spans the entire channel depth, parameterization is easy!

Experiments

• Experiments run in a 2.3 m long tank using salt water of various densities

Further details: Tan et al., Environ. Fluid Mech., 11 (2011), Tan M.Sc. thesis (U. Alberta) 2010

Sharp interfaceAmbient continuously stratified

0 0.2 0.4 0.6 0.8 1Normalized interface thickness

0

0.1

0.2

0.3

0.4

0.5

0.6

Nor

mal

ized

fron

t spe

edExperiments

• One of the challenges of running laboratory experiments is that it is difficult to minimize the thickness of the ambient interface

• Fortunately, this thickness has a very minor impact on the front speed c.f. Faust & Plate (1984)

Numerical simulations• 2D simulations (mixed spectral-FD) use Diablo (http://numerical-

renaissance.com/), which has been applied in numerous related studies e.g. Taylor 2008, Bolster et al. 2008, Flynn et al. 2008

Experiments Simulations

Further details: Tan et al., Environ. Fluid Mech., 11 (2011); Flynn, Ungarish & Tan, Phys. Fluids, 24 (2012)

Comparison (theory vs. measurement)

Fr =U

√

g′02

H

g002 = g

✓�0 � �2

�0

◆etc.

Closed symbols: expts.Open symbols: numerics

Thin lower layer Intermediate lower layer Thick lower layer

Comparison (theory vs. measurement)

Closed symbols: expts.Open symbols: numerics

Thin lower layer Intermediate lower layer Thick lower layer

• Generally positive agreement, but...

• Analytical solution “peters out” at g′12/g′02 = 0.75 Q? Why is this?

Figures from Tan et al., Environ. Fluid Mech., 11 (2011)

Gravity current front speed

Long wave/bore speed

Supercritical : g′12/g′02 < 0.75Subcritical : g′12/g′02 > 0.75

Boundary gravity current - theory

Supercritical : g′12/g′02 < 0.75

Boundary gravity current - theory

Whena qualitative change of behavior, i.e. gravity current goes from being supercritical to subcritical

g012/g002 ' 0.75 there is

Subcritical : g′12/g′02 > 0.75

Supercritical : g′12/g′02 < 0.75Subcritical : g′12/g′02 > 0.75

Super- vs. subcritical

Gravity current quickly overtaken by interfacial wave, which leads to sudden deceleration

Gravity current able to travel for long distances at constant speed

Supercritical : g′12/g′02 < 0.75Subcritical : g′12/g′02 > 0.75

Super- vs. subcritical

Gravity current quickly overtaken by interfacial wave, which leads to sudden deceleration

How quick is “quick?”

Gravity current able to travel for long distances at constant speed

Super- vs. subcriticalHorizontal distance (normalized by lock length) where gravity current front first begins to decelerate

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

2

4

6

8

10

12

g’12/g’02

X/l

0.7500.5000.375

Symbols correspond to different lower layer depths (normalized by total channel depth)

• When lower layer is thin and is small, front will travel at constant speed for a long time (c.f. Sutherland & Nault 2004)

• Not so when lower layer is thick and/or is large

g012/g002

g012/g002

Lock condition• When lower layer is thin and is small, front will travel at constant

speed for a long timeg012/g

002

g012/g002

These (generic) statements are independent of the initial (i.e. lock) condition, but not so the quantitative details of our previous parameterization

η

h′

1

η = 1

2(H − h′

1)

H

• Not so when lower layer is thick or is large

Equation applies only when the gravity current fluid initially spans the entire channel depth (full depth lock release)

Partial depth lock release

Two alternatives:

*Tan M.Sc. thesis (U. Alberta) 2010

• Generalize previous parameterization (hard problem*)

• Apply a shallow-water model

Partial depth lock release

Two alternatives:

• Generalize previous parameterization (hard problem*)

• Apply a shallow-water model

*Tan M.Sc. thesis (U. Alberta) 2010

• Away from the front, pressures are hydrostatic

• The return (i.e. right to left) flow in either ambient layer is neglected

Assumptions:

Shallow water models do not faithfully reproduce the details of the gravity current shape, but they have an impressive record of predicting the front speed (Ungarish 2009). Do they work well here? Yes!

Shallow water model (results)

0 0.5 10

0.20.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

ϕ

0 0.5 10

0.20.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

ϕ

0 0.5 10

0.20.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

ϕ

0 0.5 10

0.20.4

0.60.8

uN

0 0.5 10

0.2

0.4

0.60.8

uN

0 0.5 10

0.2

0.4

0.60.8

ϕ

uN

Normalized front speed vs. normalized lower layer depth for different lock conditions and ambient stratification

g012g002

= 0.25

g012g002

= 0.5

g012g002

= 0.75

Flynn, Ungarish & Tan, Phys. Fluids, 24 (2012)

Shallow water model (results)

0 0.5 10

0.20.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

ϕ

0 0.5 10

0.20.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

ϕ

0 0.5 10

0.20.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

0 0.5 10

0.2

0.4

0.60.8

ϕ

0 0.5 10

0.20.4

0.60.8

uN

0 0.5 10

0.2

0.4

0.60.8

uN

0 0.5 10

0.2

0.4

0.60.8

ϕ

uN

Normalized front speed vs. normalized lower layer depth for different lock conditions and ambient stratification

Column 1:

Deepest ambient (i.e. lock fluid spans 1/4 the channel depth)

Column 4:

Shallowest ambient (i.e. lock fluid spans entire channel depth)

Flynn, Ungarish & Tan, Phys. Fluids, 24 (2012)

Shallow water model (extensions)

rh1R

H

ρ1

ρ2 z

ρc

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

g’12/g’02

uN

Previous formulation can readily (and more or less successfully) be extended to an axisymmetric geometry

Open circles: Laboratory experiments

Closed circles: Shallow water simulations

Line: Shallow water theory (assumes constant front speed) Sahuri et al., Env. Fluid

Mech., 15 (2015)

Outlook/conclusions

Discharged effluent

Warm upper layer

Cold lower layer

(a)

Cold lower layer

Internal wave

Discharged effluent(b)

Warm upper layer

currentgravity

IntrusiveOcean Sill

FrontRiver water

Brackish water

z

Density

Investigation motivated by a desire to improve the understanding of the dynamics of discharge and ventilation flows in marine environments

Q? Have we been successful in this respect?

Q? Have we been successful in this respect?Ans. Yes...

• Have a deeper understanding of the interplay between the gravity current and the interfacial waves or disturbances that it may excite when the ambient is stratified

• Have developed well-corroborated analytical models for boundary gravity currents that consider different ambient and initial conditions

... but more work remains to be completed.

• Gravity currents are assumed to be compositional, so cannot readily describe flow physics of e.g. river plumes, which carry suspended sediment

• Have considered an idealized 2D geometry with a flat bottom boundary (see talk by Mitch Nicholson later in this session)

• Have ignored “long time” behavior where deceleration of the front must be considered

Outlook/conclusions

Selected publications• Sahuri, R.M., Kaminski, A.K, MRF and M. Ungarish 2015: Axisymmetric gravity currents in two-

layer density-stratified media. Env. Fluid Mech., 15, 1035-1051.

• MRF, Ungarish, M. and A.W. Tan 2012: Gravity currents in a two-layer stratified ambient: the theory for the steady-state (front condition) and lock-released flows, and experimental confirmations. Phys. Fluids, 24, 026601.

• Tan, A.W., Nobes, D.S, Fleck, B.A. and MRF, 2011: Gravity currents in two-layer stratified media. Environ. Fluid Mech. 11(2), 203-224.

• MRF, 2010: Review of An Introduction to Gravity Currents and Intrusions by M. Ungarish. J. Fluid Mech., 649, 537-539.

• MRF, Boubarne, T. and P.F. Linden, 2008: The dynamics of steady, partial-depth intrusive gravity currents. Atmosphere-Ocean, 46, 421-432.

• MRF and P.F. Linden, 2006: Intrusive gravity currents. J. Fluid Mech., 568, 193-202.

• MRF and B.R. Sutherland, 2004: Intrusive gravity currents and internal gravity wave generation in stratified fluid. J. Fluid Mech., 514, 355-383.

• Sutherland, B.R., Kyba, P.J. and MRF, 2004: Intrusive gravity currents in two-layer fluids. J. Fluid Mech., 514, 327-353.