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Damping of Turbulence by Suspended Sediment: Ramifications of
Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
• Richardson number control of saturated suspension
• Under-saturated (weakly stratified) sediment suspensions
• Critically saturated (Ricr-controlled) sediment suspensions
• Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012
When strong currents are present, mud remains turbulent and in suspension at a
concentration that gives Ri ≈ Ricr ≈ 1/4:
Sediment gradient Richardson number
Gradient Richardson
=
Number (Ri)
c < 0.3 g/l
density stratification
velocity shear
Shear instabilities occur for Ri < Ricr
“
“ suppressed for Ri > Ricr
Ri =
c > 0.3 g/l
Stratification
Shear
0.25
g = accel. of gravity
s = (s - )/
c = sediment mass conc.
s = sediment density
For c > ~ 300 mg/liter
Sediment concentration (grams/liter)
Ri ≈ Ricr ≈ O(1/4)
Amazon Shelf (Trowbridge & Kineke, 1994)
1/18
Are there simple, physically-based relations to predict c and du/dz related to Ri?
Large supply of easily suspended sediment creates negative feedback:
Gradient Richardson
=
Number (Ri)
density stratification
velocity shear
Ri > Ric
Ri = Ric
Sediment concentration
Height above bed
(b)
(a)
Height above bed
Shear instabilities occur for Ri < Ricr
“
“ suppressed for Ri > Ricr
Ri = Ric
Ri < Ric
Sediment concentration
(a) If excess sediment enters bottom boundary layer or bottom stress decreases, Ri
beyond Ric, critically damping turbulence. Sediment settles out of boundary layer.
Stratification is reduced and Ri returns to Ric.
(b) If excess sediment settles out of boundary layer or bottom stress increases, Ri
below Ric and turbulence intensifies. Sediment re-enters base of boundary layer.
Stratification is increased in lower boundary layer and Ri returns to Ric.
2/18
Height above bed
Consider Three Basic Types of Suspensions
Ri > Ricr
3) Over-saturated -- Settling limited
Ri < Ricr
1) Under-saturated
-- Supply limited
Ri = Ricr
2) Critically
saturated load
Sediment concentration
3/18
Height above bed
Consider Three Basic Types of Suspensions
Ri > Ricr
3) Over-saturated -- Settling limited
Ri < Ricr
1) Under-saturated
-- Supply limited
Ri = Ricr
2) Critically
saturated load
Sediment concentration
3/18
Damping of Turbulence by Suspended Sediment: Ramifications of
Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
• Richardson number control of saturated suspension
• Under-saturated (weakly stratified) sediment suspensions
• Critically saturated (Ricr-controlled) sediment suspensions
• Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012
Dimensionless analysis of bottom boundary
Variables
layer in the absence of stratification:
du/dz, z, h, n, u*
æ n zö
z du
= fç , ÷
u* dz
è zu* h ø
h = thickness of boundary layer or water depth, n = kinematic viscosity, u* = (tb/)1/2 = shear velocity
Boundary layer - current
log
layer
z du
= f (z / h)
z/h << 1
u* dz
z du 1
=
u* dz k
æzö
u = log ç ÷
k
è z0 ø
u*
“Overlap”
layer
n/(zu*)
<< 1
z du
= f (zu* / n )
u* dz
zo = hydraulic roughness
(Wright, 1995)
4/18
Dimensionless analysis of overlap layer with
(sediment-induced) stratification:
Additional variable
b = Turbulent buoyancy flux
b=
Height above the bed, z
u(z)
gs <c'w'>
rs
s = (s – )/ ≈ 1.6
c = sediment mass conc.
w = vertical fluid vel.
k z du
u* dz
=1
æ bk z ö
= fç 3 ÷
u* dz
è u* ø
k z du
Dimensionless ratio
bk z
º V = “stability
3
u*
parameter”
5/18
Deriving impact of z on structure of overlap (a.k.a. “log” or “wall”) layer
æ bk z ö
= fç 3 ÷
u* dz
è u* ø
k z du
k z du
u* dz
bk z
º V = “stability
3
u*
parameter”
= f (V )
Rewrite f(z) as Taylor expansion around z = 0:
k z du
2
2
df
V
d
f
= f (V ) = f
+V
+
+...
2
V
=
0
u* dz
dV V = 0 2 dV V = 0
≈0
=1
k z du
u* dz
=1+ a V
From atmospheric
studies, a ≈ 4 - 5
≈0
=a
é
ù
z
æV ö ú
u* ê æ z ö
u = êlog ç ÷ + a ò ç ÷ dzú
èzø
k
z
êë è 0 ø
úû
z0
If there is stratification (z > 0) then u(z) increases
faster with z than homogeneous case.
6/18
Eq. (1)
bk z
ºV
3
u*
= “stability
parameter”
(i)
well-mixed
-- Case (i): No stratification near the bed (z = 0 at z = z0).
Stratification effects and z increase with increased z.
-- Eq. (1) gives u increasing faster and faster with z
relative to classic well-mixed log-layer.
(e.g., halocline being mixed away from below)
-- Case (ii): Stratified near the bed (z > 0 at z = z0).
Stratification effects and z decrease with increased z.
-- Eq. (1) gives u initially increasing faster than u, but
then matching du/dz from neutral log-layer.
(e.g., fluid mud being entrained by wind-driven flow)
-- Case (iii): uniform z with z. Eq (1) integrates to
æzö
u = (1+ az )log ç ÷
k
è z0 ø
u*
stratified
z
Log elevation of height above bed
é
zæ ö ù
æ
ö
u ê
z
V
ú
u = * êlog ç ÷ + a ò ç ÷ dzú
èzø
k
z
êë è 0 ø
úû
z0
z0
(ii)
well-mixed
stratified
z
as z
z0
(iii)
well-mixed
-- u remains logarithmic, but shear is increased
buy a factor of (1+az)
z0
(Friedrichs et al, 2000)
as z
stratified
z is constant in z
Current Speed
7/18
Stability parameter, z, can be related to shape of concentration profile, c(z):
z = const. in z if
Fit a general power-law to c(z) of the form
Then
V ~ z
c ~ z
c ~ z
(Friedrichs et al, 2000)
-1
(i)
well-mixed
-A
stratified
A<1
(1-A)
z
If A < 1, c decreases more slowly than z-1
z increases with z,
stability increases upward,
u is more concave-down than log(z)
If A > 1, c increases more quickly than z-1
z decreases with z, stability
becomes less pronounced upward,
u is more concave-up than log(z)
If A = 1, c ~ z-1
z is constant with elevation
stability is uniform in z,
u follows log(z) profile
Log elevation of height above bed
æ gswsk ö
V = ç
c z
3 ÷
è rsu* ø
as z
z0
(ii)
well-mixed
stratified
A>1
z
as z
z0
(iii)
well-mixed
stratified
A=1
z is constant in z
z-A
If suspended sediment concentration, C ~
Then A <,>,= 1 determines shape of u profile
z0
Current Speed
(7/18)
If suspended sediment concentration, C ~ z-A
A < 1 predicts u more concave-down than log(z)
A > 1 predicts u more concave-up than log(z)
A = 1 predicts u will follow log(z)
Testing this relationship using
observations from bottom
boundary layers:
STATAFORM mid-shelf site,
Northern California, USA
Eckernförde Bay,
Baltic Coast,
Germany
(Friedrichs & Wright, 1997;
Friedrichs et al, 2000)
Inner shelf,
Louisiana
USA
8/18
If suspended sediment concentration, C ~ z-A
A < 1 predicts u more convex-up than log(z)
A > 1 predicts u more concave-up than log(z)
A = 1 predicts u will follow log(z)
Inner shelf, Louisiana, USA,
1993
STATAFORM mid-shelf site,
Northern California, USA,
1995, 1996
A ≈ 1.0
A ≈ 3.1
A ≈ 0.73
A ≈ 0.35
A ≈ 0.11
-- Smallest values of A < 1 are associated with concave-downward velocities on log-plot.
-- Largest value of A > 1 is associated with concave-upward velocities on log-plot.
-- Intermediate values of A ≈ 1 are associated with straightest velocity profiles on log-plot.
9/18
Normalized log of sensor height above bed
Observations showing effect of concentration exponent A on shape of velocity profile
Normalized burst-averaged current speed
Observations also show: A < 1, concave-down velocity
A > 1,
concave-up velocity
A ~ 1, straight velocity profile
(Friedrichs et al, 2000)
10/18
Damping of Turbulence by Suspended Sediment: Ramifications of
Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
• Richardson number control of saturated suspension
• Under-saturated (weakly stratified) sediment suspensions
• Critically saturated (Ricr-controlled) sediment suspensions
• Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012
Relate stability parameter, z, to Richardson number:
Ri = -
Definition of gradient Richardson number
associated with suspended sediment:
Original definition and application of z:
Relation for eddy viscosity:
Definition of eddy diffusivity:
V =
Az =
gs <c'w'> k z
rsu*3
k z du
u* dz
=1+ a V
k u* z
(1+ a V )
- <c'w'> = K z
Assume momentum and mass are mixed similarly:
Combine all these and you get:
gs(dc / dz)
rs (du / dz)2
dc
dz
Az = Kz
V
Ri =
1+ aV
So a constant z with height also leads to a constant Ri with height.
Also, if z increases (or decreases) with height Ri correspondingly increases (or decreases).
11/18
Ri =
z and Ri const. in z if
Define
c ~ z
-A
then
V
1+ aV
(i)
-1
c ~ z
V ~ z(1-A)
well-mixed
stratified
A<1
z
If A < 1, c decreases more slowly than z-1
z and Ri increase with z,
stability increases upward,
u is more concave-down than log(z)
z-1
If A > 1, c decreases more quickly than
z and Ri decrease with z, stability
becomes less pronounced upward,
u is more concave-up than log(z)
If A = 1, c ~ z-1
z and Ri are constant with elevation
stability is uniform in z,
u follows log(z) profile
Log elevation of height above bed
æ gswsk ö
V = ç
c z
3 ÷
è rsu* ø
as z
z0
(ii)
well-mixed
stratified
z
A>1
and Ri
as z
z0
(iii)
well-mixed
stratified
A=1
z and Ri are constant in z
z-A
If suspended sediment concentration, C ~
then A <,>,= 1 determines shape of u profile
and also the vertical trend in z and Ri
and Ri
z0
(Friedrichs et al, 2000)
Current Speed
(7/18)
Height above bed
Now focus on the case where Ri = Ricr (so Ri is constant in z over “log” layer)
Ri > Ricr
3) Over-saturated -- Settling limited
Ri < Ricr
1) Under-saturated
-- Supply limited
Ri = Ricr
2) Critically
saturated load
Sediment concentration
(3/18)
Connection between structure of sediment settling velocity to structure of “log-layer”
when Ri = Ricr in z (and therefore z is constant in z too).
Rouse Balance:
wsC = K z
Kz =
Earlier relation for eddy viscosity:
Eliminate Kz and integrate in z to get
But we already know
So
ws (1+ a V )
=1
ku
*
dc
dz
é
ù
w
(1+
a
V
)
ê s
ú
æ
ö
C
z ê ku
ú
* û
= çç ÷÷ ë
Cref è zref ø
c ~ z-1
and
k u* z
(1+ a V )
when Ri = const.
1+ a V =
ku
*
ws
when Ri = Ricr
12/18
1+ a V =
ku
*
ws
when Ri = Ricr . This also means that when Ri = Ricr :
V
Ri =
1+ aV
Az = K z =
k u* z
(1+ a V )
du u*
= (1+ a V )
dz k z
æzö
u*
u = (1+ az )log ç ÷
k
è z0 ø
gs(dc / dz)
Ri = rs (du / dz)2
Ricr =
ws V
ku
*
Az = Kz = ws z
du u*2
=
dz ws z
æzö
u*2
u = log ç ÷
ws
è z0 ø
Ricr rs æ u*2 ö -1
c=
ç ÷ z
gs è ws ø
2
13/18
STATAFORM mid-shelf site,
Northern California, USA
Mid-shelf site off
Waiapu River, New Zealand
(Wright, Friedrichs et al., 1999;
Maa, Friedrichs, et al., 2010)
14/18
10 1
(a) Eel shelf, 60 m depth, winter 1995-96
(Wright, Friedrichs, et al. 1999)
Sediment gradient Richardson number
10
Ricr = 1/4
0
10 -1
10 -2
0
10 - 40 cm
40 - 70 cm
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(b) Waiapu shelf, NZ, 40 m depth, winter 2004
(Ma, Friedrichs, et al. in 2008)
Ricr = 1/4
18 - 40 cm
Velocity shear du/dz (1/sec)
15/18
Application of Ricr log-layer equations fo Eel shelf, 60 m depth, winter 1995-96
æzö
u*2
u = log ç ÷
ws
è z0 ø
Ricr rs æ u*2 ö -1
c=
ç ÷ z
gs è ws ø
2
(Souza & Friedrichs, 2005)
16/18
Damping of Turbulence by Suspended Sediment: Ramifications of
Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
• Richardson number control of saturated suspension
• Under-saturated (weakly stratified) sediment suspensions
• Critically saturated (Ricr-controlled) sediment suspensions
• Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012
Height above bed
Now also consider over-saturated cases:
Ri > Ricr
3) Over-saturated -- Settling limited
Ri < Ricr
1) Under-saturated
-- Supply limited
Ri = Ricr
2) Critically
saturated load
Sediment concentration
(3/18)
(Mehta & McAnally,
2008)
More Settling
Starting at around 5 - 8 grams/liter, the return flow of water around settling flocs creates so much drag on
neighboring flocs that ws starts to decrease with additional increases in concentration.
At ~ 10 g/l, ws decreases so much with increased C that the rate of settling flux decreases with further increases
in C. This is “hindered settling” and can cause a strong lutecline to form.
Hindered settling below a lutecline defines “fluid mud”. Fluid mud has concentrations from about 10 g/l to 250
g/l. The upper limit on fluid mud depends on shear. It is when “gelling” occurs such that the mud can support a
vertical load without flowing sideways.
17/18
(Winterwerp, 2011)
-- 1-DV k-e model based on
components of Delft 3D
-- Sediment in density formulation
-- Flocculation model
-- Hindered settling model
18/18
Damping of Turbulence by Suspended Sediment: Ramifications of
Under-Saturated, Critically-Saturated, and Over-Saturated Conditions
Carl Friedrichs
Virginia Institute of Marine Science
Outline of Presentation:
• Richardson number control of saturated suspension
• Under-saturated (weakly stratified) sediment suspensions
• Critically saturated (Ricr-controlled) sediment suspensions
• Hindered settling, over-saturation, and collapse of turbulence
Presented at PECS
New York, USA, 14 August 2012