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.