3 Spatial Averaging

3-D equations of motionScaling => simplificationsSpatial averagingShear dispersionMagnitudes/time scales of diffusion/dispersionExamples

Navier-Stokes EqnsConservative form momentum eqn; x-component only

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

−=−∂∂

+∂∂

+∂∂

+∂∂

2

2

2

2

2

22 1)()()(

zu

yu

xu

xpfvuw

zuv

yu

xtu υ

ρ

1 2 3 4 5

1 “ storage” or local acceleration

2 advective acceleration

3 Coriolis acceleration [f = Coriolis param = 2ωsin(θ), θ = latitude]

4 pressure gradient

5 viscous stress

Turbulent Reynolds Eqns

( )( )[ ]

⎟⎠⎞

⎜⎝⎛ −

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛ −

∂∂

∂∂

+∂∂

−=

−∂∂

+∂∂

+∂∂

+∂∂

∂∂

+∂∂

+∂∂

∂∂

+∂∂

=++∂∂

=+=

'''''1

)()()(

'''

)averagetime(etc,'

2

2

22

wuzu

zvu

yu

yu

xu

xxp

vfwuz

vuy

uxt

uzw

yv

xu

ux

ux

uuuux

uuuu

υυυρ

Insert & time average (over bar)

Continuity

e.g., x-component; term 2

x-momentum

1 2a 3

4 5 2b

Specific termsTerm 2a: could subtract times continuity eqnu

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

=zw

yv

xuu

xuw

xuv

xuu

∂∂

+∂∂

+∂∂ NC form of momentum eqn (term 2)

Terms 5 + 2b

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

≅−≅=−∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

≅−≅=−∂∂

∂∂

≅−≅=−∂∂

xw

zuwuwu

zu

xv

yuvuvu

yu

xuuu

xu

xzxz

xyxy

xxxx

ετυ

ετυ

ετυ

''''

''''

2'' 22

x-comp of turbulent shear stress

εxx, εxy, εxz are turbulent (eddy) kinematic viscosities resulting from closure model (like Exx, Exy, Exz)

Specific terms, cont’dSimilar eqns for y and z except z has gravity.

For nearly horizontal flow (w ~ 0), z-mom => hydrostatic

gzp−

∂∂

−=ρ10

4 6z=η(x,y,t)

z

zη

∫+= ρz

a gdzpp

Use in x and y eqns

Specific terms, cont’d

Term 4

gdzx

gxx

p

gdzpxx

p

zs

a

za

∫

∫

∂∂

+∂∂

+∂∂

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

∂∂

=∂∂

η

η

ρρη

ρz=η(x,y,t) z

z4a 4b 4c

4a atmospheric pressure gradient (often negligible)

4b barotropic pressure gradient (barotropic => ρ = ρs = const)

4c baroclinic pressure gradient (baroclinic => density gradients; often negligible)

SimplificationNeglect εxx; εxy = εh; εzx = εz

Neglect all pressure terms except barotropicAnd drop over bars

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

−=−∂∂

+∂∂

+∂∂

+∂∂

zu

zyu

yxgfvuw

zuv

yu

xtu

zh εεη)()()( 2

1 2a 3 4b 2b1 2b2

Contrast with mass transport eqn

∑±⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

+∂∂

+∂∂

+∂∂ r

zcE

zycE

yycE

xwc

zvc

yuc

xtc

zyx)()()(

1 2a 2b1 2b2 7

Pressure gradient (4) in momentum eqn => viscosity (2b) not always important to balance advection (2a); depends on shear (separation). For mass transport, diffusivity (2b) always needed to balance advection (2a)

Comments3D models include continuity + three components of momentum eq (z may be hydrostatic approx) + n mass transport eqnsAbove are primitive eqs (u, v, w); sometimes different form, but physics should be sameSometimes further simplificationsSpatial averaging => reduced dimensions

Further possible simplifications

Neglect terms 2a and 2b1

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+∂∂

−=−+∂∂

zu

zxgfv

tu

zεη Linear shallow

water wave eqn

Also neglect term 1

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+∂∂

−=−zu

zxgfv zε

η Steady Ekman flow

Also neglect term 2b2Geostrophic flow

xgfv∂∂

−=−η

z-vertical

Spatial averaging y-lateral

3-D equations

φ(x,y,z,t) oceanx-longitudinal

2-D lateral average

φ(x,z,t) long reservoir; deep estuary/fjord

2-D vertical averageφ(x,y,t) shallow coastal;

estuary

1-D vertical & lateral average

φ(x, t) river; narrow/shallow estuary

1-D horizontal averageφ(z, t) deep lake/reservoir

ocean

Comments

Models of reduced dimension achieved by spatial averaging or direct formulation (advantages of both)Demonstration of vertical averaging (integrate over depth then divide by depth, leading to 2D depth-averaged models)Discussion of cross-sectional averaging (river models)

Vertical Integration => 2D (depth-averaged) eqns

z

H(x,y,t)

h(x,y,t)

η(x,y,t)

y

),,,("),,(),,,(),,,("),,(),,,(tzyxutyxUtzyxu

tzyxutyxutzyxu

h +=+=x

(notes use)

Vertical Integration, cont’d

"uu

z

H

η

-h

"cc

∫−

=η

h

dztzyxuH

tyxu ),,,(1),,(

In analogy with Reynolds averaging, decompose velocities and

concentrations into , , etc. and spatially average"uu + "cc +

Depth-averaged eqnsContinuity

0)()( =∂∂

+∂∂

+∂∂ Hv

yHu

xtη

yv

xu

zw

∂∂

∂∂

+∂∂ ,ofbcsurfacekinematicfrom

Mass and Momentum straight forward except for NL terms

( )( )

( )

( ) ( )Hcux

Hcux

dzucx

Hux

Hux

dzux

dzuudzux

dzuuuux

dzux

h

h

"")(

"

""2"")(

22

222

∂∂

+∂∂

=∂∂

⎟⎠⎞⎜

⎝⎛

∂∂

+∂∂

=

∂∂

++∂∂

=++∂∂

=∂∂

∫

∫∫∫∫∫

−

−

η

η

momentum dispersion

mass dispersion

Depth-averaged eqns, cont’dx-momentum

( ) ( ) ( )

ρτ

ρτ

εε

η

bxsxTL x

vHyx

uHx

xgHHvfHvu

yHu

xHu

t

−+⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+

∂∂

−=−∂∂

+∂∂

+∂∂ 2

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

∂∂

∂∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

∂∂

∂∂ Hvu

yuH

yHu

xuH

x yx """2 εε

Depth-avelong. diff

Long. dispersion

Depth integrated eqns, cont’dMass Transport

( ) ( ) ( )

bz

szTL z

cEzcE

ycHE

yxcHE

x

Hcvy

Hcux

Hct

∂∂

−∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+

=∂∂

+∂∂

+∂∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

∂∂

∂∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

∂∂

∂∂ Hcv

ycEH

yHcu

xcEH

x yx """"

Depth-avelong. diff

Long dispersion

Comments

εL, εT, EL , ET are longitudinal and transverse momentum and mass shear viscosity/dispersion coefficients.

εL >> εT and EL >> ET , but relative importance depends on longitudinal gradients

Dispersion process represented as Fickian(explained shortly)

Boundary Conditions: momentum

wwwDsx UVUC 22 +=ρτ Surface shear stress due to wind

(components Uw and Vw); external input (unless coupled air-water model)

Assumes Uw >> us; CD = drag coefficient ~ 10-3 (more in Ch 8)

2*

22

u

uvuC fbx

≅

+=ρτ Bottom shear stress caused by flow

(computed by model)

Different models for Cf (Darcy-Weisbach f; Manning n; Chezy C),

e.g., 2

8uf

bx ρτ =

Boundary Conditions: mass transport

sz z

cE∂∂ Surface mass transfer

(air-water exchange)z

c

0=∂∂

szc 0>

∂∂

szc 0<

∂∂

szc

No flux (dye, salt)

Source (DO)

Sink (VOC)

Boundary Conditions: mass transportBenthic mass transfer

(sediment-water exchange)bz z

cE∂∂

−

z

c

0=∂∂

−bz

c0>

∂∂

−bz

c0<

∂∂

−bz

c

No flux (dye, salt)

Source (pore water diffusion)

Sink (trace metals bound by anoxic sediments)

Magnitude of terms: Ez

uufufu

HuLuE

b

z

05.002.0,8

/velocityshear

~'~

**

*

==>≅=== ρτ

S

A∆x

HradiushydraulicpAR

gHSuSgRp

ASgu

xpuxpFSgxAF

H

H

bfg

≅==

≅==

∆=∆==∆=

/

;; *2

*

2*ρτρ

u

Normal flow; gravity balances friction

Ez cont’d

Hxuuifu

uHux

uH

HuH

E

H

HuE

vmvmvm

z

vm

z

15020;7

707.05.05.0

07.0

**

**

22

*

≅=≅=

≅≅≅

=

τ

τ

Seen previously; from analogy of mass and momentum conservation (Reynolds’analogy) and log profile for velocity

Transverse mixing: ET

24.008.0*

−≅Hu

ET Laboratory rectangular channels

(say 0.15)

6.42.0*

−≅Hu

ET

(say 0.6)

Real channels (irregularities, braiding, secondary circulation)

HB

uuifuHu

Bx

HuB

EB

tm

Ttm

2

**

2*

22

17

20;6.05.0

6.05.05.0

=

==

=≅τ B = channel width

ExampleB = 100 m, H = 5m, u = 1 m/s

Xvm = 150H = 750 m

xtm = 17B2/H = (17)(100)2/5 = 34,000 m (34 km)

It may take quite a while before concentrations can be considered laterally (transversally) uniform

SimplificationsSteady state; depth-averaged; no lateral advection or long dispersion; no boundary fluxes

( ) ( ) ( )b

zs

zTL zcE

zcE

ycHE

yxcHE

xHcv

yHcu

xHc

t ∂∂

−∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

=∂∂

+∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

ycHE

yxcuH T

Simple diffusion equation: solutions for continuous source at x=y=0

2

2

yc

uE

xc T

∂∂

=∂∂

Uniform channel

y

const B ux

Holly and Jirka (1986)

Ttm E

uBx25.0

≅

Note:

Figure by MIT OCW.

0

00

0.4

0.8

1.2

1.6

2.0

0.0001 0.001

0.2 0.4 0.6

0.01

1=cd

qd = 0

qσ = 1

0.1 1

0.2

q d -

dim

ensi

onle

ss c

umul

ativ

e di

scha

rge

c d -

dim

ensi

onle

ss c

once

ntra

tion

xd - dimensionless longitudinal distance

0.4

0.6

0.8

1.0

2BuxET

By

mHucB

&

2BuxET

0.001

0.01

0.1

31030

0.3

0.7

0.90.99

1.3

1.1

1.01

A useful extension: cumulative discharge approach (Yotsukura & Sayre, 1976)

Use cumulative discharge (Qc) instead of y as lateral variable

QQ

uH

Qc

⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

=∂∂

= ∫

cT

c

y

c

QcEuH

Qxc

dyyuyHQ

2

0

')'()'(

Dy

D behaves mathematically like diffusion coefficient, but has different dimensions; can be approximated as constant (cross-sectional average):

cT

Q

dQEuHQ

c

∫=0

21D 2

2

cQc

xc

∂∂

=∂∂

=> D Can use previous analysis

Longitudinal Shear DispersionWhy is longitudinal dispersion Fickian?

Original analysis by Taylor (1953, 1954) for flow in pipes; following for 2D flow after Elder (1959)

z

u u”

Longitudinal Shear dispersionWhy is longitudinal dispersion Fickian?

Original analysis by Taylor (1953, 1954) for flow in pipes; following for 2D flow after Elder (1959)

z

u u”H

t

to t1 t2

c L2

t

Shear dispersion, cont’dz

u u”

t

to t1 t2

H

L2

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎠

⎞⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

+∂∂

+∂∂

+∂∂

=+=+=+=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

+∂∂

zcE

zzcE

zcEcEcucucc

ttuxcccuuuycE

yxcE

xxczu

tc

zzxx

yx

""""""

;;";"

)(

ζζζζζζττ

τζ

1 2 3 4 5 6 7

Shear dispersion, cont’d

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

+∂∂

+∂∂

+∂∂

zcE

zcEcEcucucc

zxx""""""

ζζζζζζττ

z

u u”

t

to t1 t2

H

L2

1 2 3 4 5 6 7

ζτζζ

∂∂<<∂∂=>>>∂∂<<∂∂=>>>

<<=>>>

")"")

verticaldispersionallongitudin)

uLxcccccb

HLa 5, 6 << 7

4 << 3

1, 2 << 3

Shear dispersion, cont’d

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

zcE

zcu z

""ζ

z

u u”

t

to t1 t2

H

L2

3 7

Differential advection balanced by transverse diffusion

""cu−xcEL ∂∂

and to show that it is ~We want

Integrate over z twice to get c”; multiply by u”; integrate again and divide by H (depth average); add minus sign

Shear dispersion, cont’d

∫∫∫∫ ∂∂

−=−=−zz

z

HH

dzdzdzuE

ucH

dzcuH

cu0000

"1"1""1""ζ

22

"uE

HIEz

hL =EL!!!

Ih = dimensionless triple integration ~ 0.07

HuEuu z ** ~;~"

EL = 5.9u*H Elder (1959) using log profile for u”(z)

EL = 10u*roTaylor (1954) turbulent pipe flow (ro= radius)

Use EL to compute σL or use measured σL to deduce EL

Comments

EL involves differential advection (u”) with transverse mixing in direction of advection gradient (Ez)EL ~ 1/Ez; perhaps counter-intuitive, but look at time scales:

∫∞

0

2 )(~ ττ dRuD22

"~ uE

HEz

L

Tc Uc2

Recall Taylor’s Theorem

A thought experimentConsider the trip on the Mass Turnpike from Boston to the NY border (~150 miles).

Assume two lanes in each direction, and that cars in left lane always travel 65 mph, while those in the right lane travel 55 mph.

At the start 50 cars in each lane have their tops painted red and a helicopter observes the “dispersion”in their position as they travel to NY

1) How does this dispersion depend on the frequency of lane changes?

2) Would dispersion increase or decrease if there were a third (middle) lane where cars traveled at 60 mph?

Thought experiment, cont’d

22

)"(~ uE

HEz

L

What are the analogs of H, Ez and u”

Thought experiment, cont’d2

2

)"(~ uEHE

zL

H ~ number of lanes

Ez ~ frequency of lane changing

u” ~ difference between average and lane-specific speed

1) Decreasing Ez increases dispersion (as long as there is some Ez)

2) Increasing lanes increases H, decreases mean square u”,

3/503

)6055()6060()6065()"(

2/502

)6055()6065()"(

2222

222

=−+−+−

=

=−+−

=

u

u (2 lanes)

(3 lanes)

If Ez is constant, net effect is increase in EL by (3/2)2(2/3) = 50%

1D (river) dispersion

∫=

+=+=

A

cdAA

c

cccuuu1

";"

Insert into GE and spatial average

Continuity

LqAuxt

A=

∂∂

+∂∂ )( qL = lateral inflow/length [L2/T]

Mass Conservation (conservative form)

LLix cqArcuAxcEA

xAuc

xAc

t++⎥⎦

⎤⎢⎣⎡ −

∂∂

∂∂

=∂∂

+∂∂ "")()(

xcAEL ∂∂

Longitudinal dispersion again

1D (river) dispersion, cont’d

NC form from conservative equation minus c times continuity

LLix cqArcuAxcEA

xAuc

xAc

t++⎥⎦

⎤⎢⎣⎡ −

∂∂

∂∂

=∂∂

+∂∂ "")()(

⎭⎬⎫

⎩⎨⎧ =

∂∂

+∂∂

⋅− LqAuxt

Ac )(

Mass Conservation (NC form)

( )ccAqr

xcAE

xAxcu

tc

LL

iL −++⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

+∂∂ 1

Note: if cL > c, c increases; if cL < c, c decreases (dilution)

EL for riversElder formula accounts for vertical shear (OK for depth averagedmodels that resolve lateral shear); here we need to parameterizelateral and vertical shear. Analysis by Fischer (1967)

B = river width

ET = transverse dispersion coefficient

Ib – ND triple integration (across A) ~ 0.07

xcu

EBIcu

Tb ∂

∂=− 2

2

"""

EL

Same form as Elder, but now time scale is B2/ET, rather than H2/Ez. ET > Ez, but B2 >> H2 => this EL is generally much larger

EL for rivers, cont’dUsing approximations for u”, ET, etc.

HuBuEL

*

22

01.0≅ Fischer (1967); useful for reasonably straight, uniform rivers and channels

*20if uu ≅

22

**

01.0 ⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛≅

HB

uu

HuEL

or

2

*

4 ⎟⎠⎞

⎜⎝⎛≅

HB

HuEL

Magnitude of terms, revisited

22

**

*

*

*

*

01.0

10

6

6.0

07.0

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛≅

≅

≅

≅

≅

HB

uu

HuE

ruE

HuE

HuE

HuE

L

o

L

L

T

z Vertical Diffusion

Transverse Diffusion in Channels

Longitudinal Dispersion (depth-averaged flow)

Longitudinal Dispersion (turbulent pipe flow)

Longitudinal Dispersion (rivers)

Previous Example, revisited

B = 100 m, H = 5m, u = 1 m/s, u* = 0.05u =0.05

/sm400)5)(05.0(

)100()1)(01.0(01.0 222

*

22

===Hu

BuEL

Lx Edtd 22 =σ => Gaussian Distribution; but only

after cross-sectional mixing

uE

BxT

tm

25.0= xtm

xtm = 34 km if point source on river bank; u B

Previous Example, revisited

B = 100 m, H = 5m, u = 1 m/s, u* = 0.05u =0.05

/sm400)5)(05.0(

)100()1)(01.0(01.0 222

*

22

===Hu

BuEL

Lx Edtd 22 =σ => Gaussian Distribution; but only

after cross-sectional mixing

uE

BxT

tm

25.0= xtm

4 times less if point source in mid-stream u B

Previous Example, revisited

B = 100 m, H = 5m, u = 1 m/s, u* = 0.05u =0.05

/sm400)5)(05.0(

)100()1)(01.0(01.0 222

*

22

===Hu

BuEL

Lx Edtd 22 =σ => Gaussian Distribution; but only

after cross-sectional mixing

uE

BxT

tm

25.0= xtm

uLess still if distributed across channel (but not zero) B

Storage zonesReal channels often have backwater (storage) zones that increase dispersion and give long tails to c(t) distribution

A

As

c(x,t)

t

Storage zones, cont’d

( )

( )ccAA

dtdc

ccccAq

xcAE

xAxcu

tc

ss

s

sLL

L

−−=

−+−+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

+∂∂

α

α )(11)

2)

Main channel

Storage zone

A(x) = cross-sectional area of main channel

As = cross-sectional area of storage zone

α = storage zone coefficient (rate, t-1; like qL/A)

If you multiply 1) by A and 2) by As, the exchange terms are αA(cs-c) and –αA(cs-c)

Really the same process as longitudinal dispersion, but instead of cars in either fast or slow lane, some are in the rest stop.