Outline of Presentation : Richardson number control of saturated suspension

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

Damping of Turbulence by Suspended Sediment: Ramifications of Under-Saturated, Critically-Saturated, and Over-Saturated Conditions

Carl FriedrichsVirginia Institute of Marine Science

Presented at PECSNew York, USA, 14 August 2012

g = accel. of gravitys = (s - )/ c = sediment mass conc.s = sediment density

Stratification

Shear=c < 0.3 g/l c > 0.3 g/l

Amazon Shelf (Trowbridge & Kineke, 1994)

Ri ≈ Ricr ≈ O(1/4)

For c > ~ 300 mg/liter Sedi

men

t gra

dien

t Ric

hard

son

num

ber

Sediment concentration (grams/liter)

0.25

When strong currents are present, mud remains turbulent and in suspension at a concentration that gives Ri ≈ Ricr ≈ 1/4:

Gradient RichardsonNumber (Ri) = density stratification

velocity shearShear instabilities occur for Ri < Ricr

“ “ suppressed for Ri > Ricr

Ri

-

1/18

(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.

Sediment concentration

Hei

ght a

bove

bed

Sediment concentrationH

eigh

t abo

ve b

ed

Ri = Ric Ri < Ric

Ri > Ric

Ri = Ric

Large supply of easily suspended sediment creates negative feedback:

Gradient RichardsonNumber (Ri) = density stratification

velocity shearShear instabilities occur for Ri < Ricr

“ “ suppressed for Ri > Ricr

(a) (b)

Are there simple, physically-based relations to predict c and du/dz related to Ri?

2/18


Heig

ht a

bove

bed

Ri = Ricr

Ri > Ricr

Ri < Ricr

Consider Three Basic Types of Suspensions

3) Over-saturated -- Settling limited

1) Under-saturated-- Supply limited

2) Critically saturated load

3/18


Heig

ht a

bove

bed

Ri = Ricr

Ri > Ricr

Ri < Ricr

Consider Three Basic Types of Suspensions




3/18






Dimensionless analysis of bottom boundary layer in the absence of stratification:

Variables du/dz, z, h, n, u*

Boundary layer - current log layer

(Wright, 1995)

zo = hydraulic roughness

“Overlap” layer

n/(zu*) << 1

z/h << 1

h = thickness of boundary layer or water depth, n = kinematic viscosity, u* = (tb/)1/2 = shear velocity

4/18

Dimensionless analysis of overlap layer with (sediment-induced) stratification:

Dimensionless ratio

= “stability parameter”

Additional variableb = Turbulent buoyancy flux

Heig

ht a

bove

the

bed,

z

u(z)

s = (s – )/ ≈ 1.6c = sediment mass conc.

w = vertical fluid vel.

5/18

Rewrite f(z) as Taylor expansion around z = 0:

= 1 = a≈ 0 ≈ 0

From atmospheric studies, a ≈ 4 - 5

If there is stratification (z > 0) then u(z) increases faster with z than homogeneous case.

Deriving impact of z on structure of overlap (a.k.a. “log” or “wall”) layer


6/18

Current Speed

Log

elev

atio

n of

hei

ght a

bove

bed

z as z

well-mixedstratified

(i)

(iii)

(ii)

z is constant in z

well-mixed stratified


z as z

-- 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)

z0

z0

z0

Eq. (1)

-- Case (iii): uniform z with z. Eq (1) integrates to

-- u remains logarithmic, but shear is increased buy a factor of (1+az)

(Friedrichs et al, 2000)


7/18

If suspended sediment concentration, C ~ z-A

Then A <,>,= 1 determines shape of u profile

(Friedrichs et al, 2000)

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 elevationstability is uniform in z,u follows log(z) profile

z = const. in z if

Fit a general power-law to c(z) of the form

Then

Current Speed

Log

elev

atio

n of

hei

ght a

bove

bed

z as z


(i)

(iii)

(ii)

z is constant in z



z0

z0

z0

z as z

A < 1

A = 1

A > 1

Stability parameter, z, can be related to shape of concentration profile, c(z):

(7/18)

STATAFORM mid-shelf site, Northern California, USA

Inner shelf, Louisiana

USA

Eckernförde Bay,Baltic Coast,

Germany


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:

(Friedrichs & Wright, 1997;Friedrichs et al, 2000)

8/18

STATAFORM mid-shelf site, Northern California, USA,

1995, 1996

Inner shelf, Louisiana, USA,1993

-- 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.

A ≈ 0.11

A ≈ 3.1

A ≈ 0.35

A ≈ 0.73

A ≈ 1.0


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)

9/18

Observations showing effect of concentration exponent A on shape of velocity profile

Normalized burst-averaged current speed

Nor

mal

ized

log

of s

enso

r hei

ght a

bove

bed

Observations also show: A < 1, concave-down velocityA > 1, concave-up velocity

A ~ 1, straight velocity profile(Friedrichs et al, 2000)

10/18






Relate stability parameter, z, to Richardson number:

Definition of gradient Richardson number associated with suspended sediment:

Original definition and application of z:

Assume momentum and mass are mixed similarly:

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).

Relation for eddy viscosity:

Definition of eddy diffusivity:

Combine all these and you get:

11/18

(Friedrichs et al, 2000) Current Speed

Log

elev

atio

n of

hei

ght a

bove

bed

z and Ri as z


(i)

(iii)

(ii)

z and Ri are constant in z



z0

z0

z0

A < 1

A = 1

A > 1z and Ri as z


then A <,>,= 1 determines shape of u profileand also the vertical trend in z and Ri

z and Ri const. in z if

thenDefine

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)

If A > 1, c decreases more quickly than z-1

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 elevationstability is uniform in z,u follows log(z) profile

(7/18)


Heig

ht a

bove

bed

Ri = Ricr

Ri > Ricr

Ri < Ricr

Now focus on the case where Ri = Ricr (so Ri is constant in z over “log” layer)




(3/18)

Eliminate Kz and integrate in z to get

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:

Earlier relation for eddy viscosity:

But we already know when Ri = const.

So and when Ri = Ricr

12/18

when Ri = Ricr . This also means that when Ri = Ricr :

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

(a) Eel shelf, 60 m depth, winter 1995-96(Wright, Friedrichs, et al. 1999)

Velocity shear du/dz (1/sec)

10 - 40 cm40 - 70 cm

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410-2

10-1

100

101

Ricr = 1/4Se

dim

ent g

radi

ent R

icha

rdso

n nu

mbe

r

(b) Waiapu shelf, NZ, 40 m depth, winter 2004(Ma, Friedrichs, et al. in 2008)

Ricr = 1/4

18 - 40 cm

15/18

Application of Ricr log-layer equations fo Eel shelf, 60 m depth, winter 1995-96

(Souza & Friedrichs, 2005)

16/18







Heig

ht a

bove

bed

Ri = Ricr

Ri > Ricr

Ri < Ricr

Now also consider over-saturated cases:




(3/18)

More Settling(Mehta & McAnally,

2008)

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






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Outline of Presentation : Richardson number control of saturated suspension