Environmental Fluid Mechanics Ι
1
Institute for Hydromechanics
MASS TRANSPORT PROCESSES
When a tracer cloud is introduced into a fluid flow (e.g. river) two processes are visible:
• The cloud is carried away from the point of discharge by the mean current.
• The cloud spreads in all directions due to irregular motions and grows in size.
Advection
-
mass transport by mean velocity field
Diffusion
-
mass transport by irregular fluctuations ("random movements")
mean
flow
t1
t2
PASSIVE MIXING PROCESSES
-
advection + diffusion as they exist
in environment (passive source)
ACTIVE MIXING PROCESSES
-
generation of mean and random
velocity field
by source - momentum or buoyancy
passive mixing
(weak source)
Active mixing
(strong source e.g. high velocity)
Environmental Fluid Mechanics Ι
2
Institute for Hydromechanics
MOLECULAR DIFFUSION
− molecular diffusion of dissolved or suspended matter is caused by random movement
through molecular motions of individual molecules
∂c
<0
∂x
window of unit area
J x = diffusive mass flux =
⎡ M ⎤
mass
=⎢ 2 ⎥
area , time ⎣ L , T ⎦
Fick's Law of Diffusion:
− rate of transport is proportional to the spatial concentration gradient
Jx = − D AB
∂c
∂x
∂c
∂x
Fick´s law (1-D)
⎡ L2 ⎤
D AB = coefficient of molecular diffusivity = ⎢ ⎥ = f (solvent, solute, temperature...)
⎣T⎦
Typical values for molecular diffusion:
D ∼ 2 × 10-5 cm2/s
D ∼ 2 × 10-1 cm2/s
Dissolved matter in water:
Gases in air:
Generalization:
r
J = J x , J y , Jz
(
⎛ ∂ ∂ ∂⎞
∇=⎜ ,
, ⎟ divergence vector
⎝ ∂ x ∂ y ∂z ⎠
)
r
J = −D∇C
Example:
Fick´s law (3-D)
Salt diffusion in a vessel
Water
Water
initial sharp
interface
two hours later
diffuse interface
~ 1,0 cm
Water + NaCl+ Dye
Water + NaCl+ Dye
Environmental Fluid Mechanics Ι
3
Institute for Hydromechanics
ADVECTIVE MASS FLUX
− transport of matter due to mean motion (current)
r
q = (u, v, w ) = hydrodynamic velocity
r ⎡ M L⎤ ⎡ M ⎤
c q = ⎢ 3 ⎥ = ⎢ 2 ⎥ = advective mass flux
⎣ L T ⎦ ⎣L , T ⎦
window of unit area
Example:
Pure advection in x-direction (no diffusion no reaction)
t1
t2
no change of shape
“plug flow”
TOTAL MASS FLUX:
Advection + diffusion
r
r r
r
N = N x , N y , Nz = c q + J = c q − D∇C
(
)
Environmental Fluid Mechanics Ι
4
Institute for Hydromechanics
MASS CONSERVATION PRINCIPLE
R = rate of production or decay per unit
⎡ M ⎤
volume = ⎢ 3 ⎥
⎣L , T ⎦
⎡net flux of mass⎤
⎢ (In − Out ) ⎥
⎣
⎦
−
+
⎡Rate of production / ⎤
⎢decay of mass
⎥
⎢
⎥
⎢(chemical, physical, ⎥
⎢
⎥
⎣bio log ical ... )
⎦
=
⎡Rate of mass ⎤
⎢accumulation ⎥
⎢
⎥
⎢⎣ within element ⎥⎦
∂ Ny
∂ Nx
∂ Nz
∂c
∆ x∆ y ∆ z −
∆ x∆ y ∆ z −
∆ x∆ y ∆ z + R ∆ x∆ y ∆ z =
∆ x∆ y ∆ z
∂x
∂y
∂z
∂t
r
r
∂ c ∂ N x ∂ N y ∂ Nz ∂ c
∂c
+
+
+
=
+ ∇⋅N =
+ ∇ ⋅ c q − ∇ ⋅ (D ∇ c) = R
∂t
∂x
∂y
∂z
∂t
∂t
∂c r
+ q ⋅ ∇ c = D ∇2 c + R
∂t
∂c
∂t
+
∂c
u
∂x
+
∂c
v
∂y
+
∂c
w
∂z
↑
1444442444443
Temporal
Advective transport
change
Convective Diffusion Equation
=
⎛ ∂2 c
∂2 c
∂2 c ⎞
D⎜ 2 +
+
⎟
∂ y2
∂ z2 ⎠
⎝∂x
14444244443
Diffusive transport
+
R
↑
Reaction
Environmental Fluid Mechanics Ι
5
Institute for Hydromechanics
Solution of the convective diffusion equation:
1. Basic solution: Instantaneous point source (IPtS) in uniform current and infinite
space
z
U
r
q = (U, 0,0) , (uniform velocity field)
R = − k c (first order decay)
y
t=0
t
1
t
2
x
⎛ ∂2 c ∂2 c ∂2 c ⎞
∂c
∂c
+U
=D⎜ 2 +
+
⎟ − kc
∂t
∂x
∂ y2 ∂ z2 ⎠
⎝∂x
Initial Conditions (I.C.)
t < 0, c = 0
t=0
Sudden mass input
M on (x1, y1, z1)
Boundary Conditions (B.C.)
c → 0 as
(x, y, z) → ∞
Solution:
[( x − x ) − Ut]
−
1
c
=
M
(4 π D t)
3/ 2
123
cmax
e
2
+ ( y − y1) + ( z − z1)
2
4D t
2
e− k t
14444424444443 123
spatial distribution
non-conservative
(reaction) effect
1. Position of cloud center:
( x1 + U t, y1, z1)
t1
2. Cloud size:
4 D t = 2 σ2 ,
σ = variance
σ = 2D t = standard deviation ∼ t
t2
0,5
cmax
Distance relative to center
3. Maximum concentration
cmax ∼ t-3/2
Environmental Fluid Mechanics Ι
6
Institute for Hydromechanics
2. Other solutions
Boundary conditions (finite space):
on ⎡:
c = specified
Ex: c = 0
Absorption condition
(Dirichlet B.C.)
∂c
= specified
∂n
Ex:
∂c
=0
∂n
Diffusion barrier (Reflection condition)
(Neumann B.C.)
Multiple sources
→
→
superposition, due to linearity of governing equation
image sources
z
k
c = ∑ ci
i=1
i =1
x
i=2
Time dependence
Initial condition:
− Instantaneous sources
− Continuous sources
Environmental Fluid Mechanics Ι
TURBULENCE
7
Institute for Hydromechanics
- an instability of a base flow (shear flow)
Possibility 1:
- damping of disturbance
(viscosity)
- stable flow
damping
LAMINAR FLOW
Possibility 2:
- insufficient damping
- large eddies
- secondary eddies
TURBULENT FLOW
disturbance
grows
SPECTRUM OF EDDIES
Large eddies: Integral length scale
=>
lΙ
Integral time scale
Integral velocity scale uΙ
tΙ =
lΙ
uΙ
⎡ L2 ⎤
=⎢ 3 ⎥
⎣T ⎦
Energy cascade (transfer of energy from larger to smaller scales)
2
Extraction of energy from mean flow: ε~
3
K.E. uΙ
u
~
= Ι
time l Ι / uΙ l Ι
Smallest eddies: at some scale (Kolmogorov length scale) the eddy size will be small
enough so that viscous damping will prevent further breakdown
⎡ L2 ⎤
Dimensionality: viscosity ν = ⎢ ⎥ ,
⎣T⎦
Kolmogorov length scale
Turbulence spectrum:
Overall criterion:
⎡ L2 ⎤
energy dissipation rate ε = ⎢ 3 ⎥
⎣T ⎦
ν3 / 4
η = 1/ 4
ε
ε
lΙ
η
" equilibrium subrange"
Reynolds number
U ∼ base velocity,
UL
> 100 to 1000
ν
L ∼ base flow length
Re =
Environmental Fluid Mechanics Ι
8
Institute for Hydromechanics
TURBULENT DIFFUSION
Random motion due to turbulent velocity fluctuations cause additional transport ("small
scale advection").
u ′ ( t) = fluctuating velocity
u (t) =
1 t −T
∫ u ( t ′) d t ′ = mean velocity
T t
u = u + u′
w = w + w′
v = v + v′
c = c + c′
u′ =
1 t −T
∫ u′ ( t ′ ) d t ′ = 0
T t
Averaging process:
Reynolds decomposition
uc = u c + uc ′ + u′ c + u′ c ′
⎤
⎛ ∂2 c ∂2 c ∂2 c ⎞
1 t + T ⎡ ∂c ∂uc ∂ vc ∂ wc
+
+
+
=
D
⎜ 2 + 2 + 2 ⎟ + R⎥ d t
∫ ⎢
∂x
∂y
∂z
T t ⎢⎣ ∂ t
∂y
∂z ⎠
⎝ ∂x
⎥⎦
⎛ ∂2 c ∂2 c ∂2 c ⎞
∂c
∂c
∂c
∂c
∂
∂
∂
+u
+v
+w
=−
u′ c′ −
v ′c ′ −
w ′ c′ + D ⎜ 2 + 2 + 2 ⎟ + R
∂t
∂x
∂y
∂z
∂x
∂y
∂z
∂y
∂c ⎠
⎝ ∂x
144424443
time
advective transport
change
by
of mean
mean velocity
conc.
14444244443
"small scale advection"
⇓
turbulent mass transport
14442443
molecular
diffusion
≈0
r
⎡ M ⎤
Jt = u′c ′, v ′c ′, w ′c ′ = turbulent diffusive mass flux = ⎢ 2 ⎥
⎣L , T ⎦
(
)
r
Jt needs parameterization
: analogy to molecular diffusion (Fick's law) ?
↓
↓
reaction
Environmental Fluid Mechanics Ι
9
Institute for Hydromechanics
Example: Tank with Grid Stirring
net transport w ′c′ > 0
DIFFUSION IN TURBULENT FLOW
→
fluctuating velocity field
⇒
multiple scale eddies (spectrum)!
Ex: Release from POINT SOURCE
σ ∼ t
σ ~t
a)
0,5
Short time after release t<t Ι , σ < l Ι
First small eddies, then eddies of increasing size will cause spreading
σ x = u′ 2 t ~ t1
Rapid growth (Taylor theorem, 1921)
u′2 =rmsvelocity~uΙ
b)
Long time after release t>t Ι :
Large eddies control spreading
( )
σ x = 2 uΙ t Ι t ~t1 / 2
2
∴
eddies are independent of each other
=> random process
"Fickian" behavior: corresponds to spreading with
gradient-type diffusion and a constant coefficient
⎡ L2 ⎤
Analogy:
uΙ t Ι =uΙ l Ι =E x = turbulentdiffusivity = ⎢ ⎥
⎣T⎦
∂c
∂c
∂c
,
,
J tx = u′ c ′ = − E x
Jty = v ′ c ′ = − E y
J tz = w ′ c ′ = − E z
∂y
∂z
∂x
2
Environmental Fluid Mechanics Ι
10
Institute for Hydromechanics
PRACTICAL TURBULENT DIFFUSION EQ.:
c
→
c
∂c
∂c
∂c
∂c
∂ ⎛ ∂ c⎞
∂ ⎛ ∂ c⎞
∂ ⎛ ∂ c⎞
+u
+v
+w
=
⎟ + R + D ∇2 c
⎜E
⎟+
⎜ Ey
⎟+
⎜ Ex
∂t
∂x
∂y
∂ z ∂ x ⎝ ∂ x⎠ ∂ y ⎝ ∂ y ⎠ ∂ z ⎝ z ∂ z⎠
14444444244444443
turbulent diffusive transport
Valid for
t > tΙ
σ > lΙ
;
Diffusivities can have dependence on spatial position and direction:
-
E x , E y , E z = f ( x , y , z)
non-homogeneous
if
E x , E y , E z = const.
homogeneous
-
Ex ≠ Ey ≠ Ez
anisotropic
if
Ex = Ey = Ez = E
isotropic
∂2c
∂2c
∂2c
+
E
+
E
y
x
∂ x2
∂y 2
∂z2
→
Ex
→
⎛ ∂2c
∂2c
∂2c ⎞
⎟
E⎜ 2 +
+
∂y 2
∂z2 ⎠
⎝ ∂x
Estimation of turbulent diffusivities:
E = f (base flow characteristics)
Order of magnitude:
E ~uΙ l Ι single scale only!
TURBULENT CHANNEL FLOW
Uh
> 500 for turbulence
ν
τ 0 = ρ gh S = shear stress
Re =
τ0
= u∗ = friction (shear ) velocity
ρ
u* = g hS
Dominating scales:
uΙ ~u∗,l Ι ~h
Also:
f
τ 0 =ρ U 2
8
Or:
1
U = h 2 / 3 S1/ 2
n
f = Darcy-Weisbach coefficient, 0.02 .. 0.1
n = Mannings's n, 0.02 ... 0.05
Rule of thumb: u ∗ ≈ ( 0.05 to 0.1) U
u∗ =
f
U²
8
u∗ =
gn 2
U²
h 1/ 3
Environmental Fluid Mechanics Ι
11
Institute for Hydromechanics
MIXING IN RIVERS
Eddy diffusivity Ez
We expect: E z ~ u ∗ h
Velocity profile
du
= f (u ∗ , z)
dz
Dimensional analysis:
du u ∗
du u ∗
~
=
or
dz z
dz κ z
Upon integration:
u
1
= lnz + C 1
u∗ κ
log-law
κ = von Karman constant = 0.4
C1 = f (Re, roughness)
du
⎛ z⎞
τ = τ 0 ⎜ 1− ⎟ = ρ ε z
⎝ h⎠
dz
model for momentum exchange
(Boussinesq)
⎛ z⎞
τ0 ⎜ 1− ⎟
⎝ h⎠
z⎞
⎛
= κ u∗ ⎜ 1 − ⎟
εz =
du
⎝
h⎠
ρ
dz
εz
Ez
z⎛ z ⎞
= κ ⎜ 1− ⎟ ≡
u∗ h
h ⎝ h ⎠ u∗ h
⎡ L2 ⎤
εz = eddy viscosity = ⎢ ⎥
⎣T⎦
Reynolds analogy for turbulent flows
Mass exchange ≈ momentum exchange
Depth averaged:
Ez
κ
= = 0.067 ≈ 0.1
u∗ h 6
Rules of thumb:
River U, h
u ∗ ≈ 0.1U
E 2 ≈ 0.1u ∗ h ≈ 0.01Uh
Ex:
U = 2 m/s, h = 2m
Ez = 0.04 m²/s = 400 cm2/s >> D = 2 × 10-5 cm/s !!
Environmental Fluid Mechanics Ι
12
Institute for Hydromechanics
VERTICAL MIXING
e.g. source at surface
σ z = 2E z t = 2E z
Complete vertical mixing: 2,15 σz = h
x
U
10% criterion with Gaussian profile
2
⎛ h ⎞ U
xm = ⎜
⎟
⎝ 2.15 ⎠ 2E z
h
=
2.15
xm
2 Ez
U
Ex:
2
⎛ 2 ⎞
xm = ⎜
= 25 m
⎟
⎝ 2.15 ⎠ 2 × 0.04
⇒
2
Very fast
Typically: Distance to complete vertical mixing 10 to 20 water depth!
LATERAL MIXING
Wide Channel, Depth h
Conditions:
x > xm
σy > h
Assume isotropy: E y ≅ Ez = 0.067 u ∗ h
Ey
≅ 0.15 to 0.20
u∗ h
Evaluation of field / laboratory tests:
Data:
2
2
2
1 d σy
1 σ y ( t 2 ) − σ y ( t1 )
= Ey ≅
2 dt
2
t 2 − t1
or
2
2
1 σ y ( x 2 ) − σ y ( x1 )
2
( x 2 − x1 ) / U
Outlines of tracer cloud
Data from Mississippi River study on
longitudinal mixing (after McQuivey and
Keefer 1976b)
straight, uniform channels
Environmental Fluid Mechanics Ι
13
Institute for Hydromechanics
NATURAL CHANNELS; IRREGULARITIES
Data:
Ey
u∗h
= 0.5 to 10
. =α
α = 0.6 ± 50%
⎛ UB ⎞
⎟
α = 0.4 ⎜
⎝ u∗ Rc ⎠
Cumulative Discharge Method
(Fischer (1972)
2
Yotsukura and Sayre (1976)
Yotsukura and Cobb (1976)
dq = hudy = local discharge → from
measurement or numerical model
y
q ( y) = ∫ hudy = cumulative discharge
0
B
Q = ∫ hud y = total discharge
0
Approximate local balance:
∂c
∂c⎞
∂ ⎛
=
u ( y) h ( y)
⎜ E y ( y) h ( y) ⎟
∂x
∂y ⎝
∂y⎠
↓
↓
depth integrated
advection
uh
depth integrated
lateral diffusion
∂c
∂ ⎛
∂c ⎞
= uh ⎜ h 2 E y u ⎟
∂x
∂q ⎝
∂q ⎠
⇒
∂c
∂c⎞
∂ ⎛ 2
=
⎜ h Ey u ⎟
∂x ∂q ⎝
∂ q⎠
14243
⎡ L5 ⎤
"Diffusitivity" = ⎢ 2 ⎥ ≈ constant
⎣T ⎦
Understanding of advective velocity
field is key to many environmental
diffusion problems!
Environmental Fluid Mechanics Ι
y/B
14
Institute for Hydromechanics
q/Q
Transverse distributions of dye observed in the Missouri River near Blair, Nebraska, by
Yotsukura et al. /1970), plotted (a) versus actual distance across the stream and (b)
versus relative cumulative discharge [Yotsukura and Sayre (1976)].
Environmental Fluid Mechanics Ι
15
LONGITUDINAL DISPERSION
(SHEAR FLOW DISPERSION)
Institute for Hydromechanics
The stretching of a tracer cloud due to vertical and horizontal velocity shear is called
longitudinal dispersion. In this case, the velocity differences interact with the transverse
diffusion effect and produce an additional transport mechanism
Observations: t > t mixing or σ z > h
For long time:
1. Gaussian distribution
2. σ L − t
Therefore: behavior is analogous to
Fickian diffusion
Analysis:
∂c
∂c
∂ ⎛ ∂c ⎞
∂ ⎛
∂c ⎞
+ u ( z, t)
=
⎜E x ⎟ +
⎜Ez
⎟
∂t
∂x ∂x ⎝ ∂z⎠ ∂z ⎝
∂z⎠
2-D
Spatial decomposition: u = U + u ′′ ,
u ′′ , c ′′ = spatial deviations
Substitute and average
c = C + c ′′ ,
1h
( ) dz
h ∫0
U, C = spatial averages
∂C
∂C ∂u ′′ c ′′
∂ ⎛ ∂C ⎞ 1 ⎛ ∂ c ⎞
+U
+
=
⎜E
⎟ + ⎜E
⎟
∂t
∂x
∂x
∂ x ⎝ x ∂ x ⎠ h ⎝ z ∂z ⎠
h
u ′′ c ′′ =
1
u ′′ c ′′ dz
h ∫0
h
0
14243
= 0 no flux
⎡ M ⎤
JL = u ′′ c ′′ = dispersive mass flux = ⎢ 2 ⎥
⎣L , T ⎦
Environmental Fluid Mechanics Ι
16
Institute for Hydromechanics
⎡ M ⎤
JL = u ′′ c ′′ = dispersive mass flux = ⎢ 2 ⎥
⎣L , T ⎦
JL = − EL
− neglect:
long. diffusion,
∂C
∂t
↓
+
U
change of
mean conc.
∂C
∂x
↓
=
advection
mean
velocity
⎡L2 ⎤
EL = coeff. of longitudinal dispersion = ⎢ ⎥
⎣T⎦
Ex << EL
∂C
∂x
Analogy:
EL
∂2 C
∂ x2
↓
1-D
spreading by
velocity deviations
+ transverse diffusion
Local balance as seen by moving observer (U):
u ′′
∂C
∂x
=
∂ ⎛
∂ c ′′ ⎞
⎜E z
⎟
∂z ⎝
∂z ⎠
↓
Taylor (1953, 1954)
↓
transport in x
direction by
differential
velocities
⇔ diffusion in
vertical direction
by turbulence
"Stretching" ⇔
With given properties: u ′′ ( z), E z ( z)
and evaluate
one can compute
u ′′ c ′′
This leads to:
h
z
z
1
1
EL = − ∫ u ′′ ∫
u ′′ dz dz dz
h 0 0 E z ∫o
c ′′ ~
"Homogenization"
∂C
,
∂x
Environmental Fluid Mechanics Ι
Applications:
17
Institute for Hydromechanics
2-D Flows
1. Channel flows:
u ′′ ~ log− profile
EL = 5.9 u∗ h
Elder (1959)
EZ ∼ parabolic
Compare:
E y ≈ E x = 0.2 u∗ h
EL >> Ex
Plan view of a drop of dye diffusing in the turbulent flow in an open channel.
Distribution of concentration C, normalized to have a maximum of 10. The flow is
to the left; h = 1.43 cm, x/h = 90 ( Elder, 1959)
2. Turbulent pipe flow:
EL = 10.1 u∗ ro
ro = pipe radius
Taylor (1954)
3. Laminar pipe flow:
EL
mol
r 2 o U2
=
48 D
General rule for applicability:
Taylor (1953)
t > tM = transverse mixing time (vertical)
= 0.4
h2
Et
Et = transverse diffusivity
Environmental Fluid Mechanics Ι
18
Data:
Natural Rivers
Institute for Hydromechanics
EL
= 100 to 1000 3-D non-uniformities
u∗ h
Lateral velocity
deviations dominate
→ Stretching
+
Lateral (transverse) diffusion
Typical cross section of a natural stream
(Fischer 1968)
Fischer (1965) u ( y), E y ( y) ,
y
B
1
1
EL = − ∫ ( u − U) h ∫
A0
Ey h
0
↓
A = cross-section
y
∫ ( u − U) h dydydy
u(y): measurements
0
↓
or numerical model
velocity deviation in lateral
direction y
-
approximate formula (typical conditions):
U2 B 2
EL = 0.011
u∗ h
good within factor 4
Applicability: t > tm =
=
(σL ∼ within 2)
transverse (lateral mixing time
(B / 2) 2
0.4
Ey
Other physical non-uniformities:
Overall effects:
- stretching of "cloud"
- increased longitudinal
dispersion EL ↑
Environmental Fluid Mechanics Ι
19
Institute for Hydromechanics
Practical longitudinal dispersion equation
∂C
∂C
∂2 C
+U
= EL
− kC
∂t
∂x
∂x2
t > tM
(often neglected; but careful!)
Instantaneous Area Source:
C=
Continuous Area source:
C=
In continuous problems
Æ long. Dispersion usually negligible
⎡ ( x − Ut) 2 ⎤
⎥ exp [ − kt]
exp − ⎢
4 π EL t
⎢⎣ 4 EL t ⎥⎦
m ′′
⎡ xU
exp ⎢
4kEL
⎢⎣ 2 EL
u 1+
2
u
q′′
Plug flow:
C=
⎛
4 kEL
⎜1 m 1 +
U2
⎝
4kEL
→0
U2
q′′
⎛ kx ⎞
exp ⎜ − ⎟
⎝ U⎠
U
⎞⎤
⎟⎥
⎠ ⎥⎦
Environmental Fluid Mechanics Ι
20
Institute for Hydromechanics
OCEANIC DIFFUSION (coastal zone, estuaries, large lakes)
-
eddy structure
RELATIVE DIFFUSION"
diffusion relative to center of mass, independent of actual location in fixed space
(x, y)
-
mass will follow larger eddies, but will be diffused by action of smaller eddies (of
increasing size)
→ accelerating growth rate
Er = C ε1/3 σr4/3
↑
cloud size
„Richardson´s (1921) 4/3 Law“
d σ 2r
= 4 Er
dt
cylindrical coordinate r,
⇒ σ 2r ~ ε t 3
rapid growth
Er ~ ε t 2
standard deviation σr
≠ const.!, diffusivity increases with time
Environmental Fluid Mechanics Ι
Data:
21
Institute for Hydromechanics
σ 2 r ~ t 2.3
Er ~ σ 11. r
Okubo (1971)
→
Oceanic Diffusion Diagrams
ocean, coastal zone
large lake, coastal zone
mid-size
small lake
A. Okubo, "Oceanic Diffusion Diagrams", Deep Sea Research, Vol. 18, 1971
Typical growth rates:
1 hr
1 day
1 week
1 month
∼
∼
∼
∼
σr = 50 m
1 km
10 km
100 km
Applications:
Instantaneous Point Source (Accident):
cmax ~
1
σ 2r
2-D growth
(vertically mixed)
c=
M
e
h 2 π σr 2
⎛ r2 ⎞
⎟
− ⎜⎜
2⎟
⎝ 2σ r ⎠
t2
t1
σr
σr
c max
c max
Environmental Fluid Mechanics Ι
22
Institute for Hydromechanics
Continuous release (Routine problems):
U
∂c
∂ ⎛ ∂ c⎞
=
⎜E
⎟ − kc
∂x ∂y ⎝ y ∂y ⎠
⎛L⎞
E y = E y0 ⎜ ⎟
⎝ b0 ⎠
Brooks (1960)
4/3
4/3 Law
L = 2 3 σy
22% width
8 E y0 x ⎤
L ⎡
= ⎢1 +
⎥
b ⎣
Ub 2 0 ⎦
c max
= erf
c0
3/2
∼ x3/2 !
3/2
3
8 E y0 x ⎞
⎛
⎜1 +
⎟ −1
Ub 2 0 ⎠
⎝
Environmental Fluid Mechanics Ι
References:
23
Institute for Hydromechanics
Diffusion and Dispersion Processes
Bird, R.B., Stewart, W.E., and Lightfoot, E.N., 1960, "Transport Phenomena", John Wiley
& Sons, New York
Carslaw, H.S. and Jaeger, J.C., 1959, "Conduction of Heat in Solids", Oxford Press, 2nd
ed.
Elder, J.W., 1959, "The Dispersion of Marked Fluid in Turbulent Shear Flow", Fluid
Mechanics, Vol. 5, Part 4, pp 544-560
Fischer, H.B., 1973, "Longitudinal Dispersion and Turbulent Mixing in Open-Channel
Flow", Annual Review of Fluid Mechanics, Vol. 5, pp 59-78, Annual Review, Inc., Paolo
Alto, Calif.
Fischer, H.B., et al., 1979, "Mixing in Inland and Coastal Waters", Academic Press, New
York
Holly, E.R. and Jirka, G.H., 1986, "Mixing and Solute Transport in Rivers", Field Manual,
U.S. Army Corps of Engineers, Waterways Experiment Station, Tech. Report E-86-11
(NTIS No. 1229110, AD-A174931/6/XAB)
Jirka, G.H. and Lee, J.H.-W., 1994, "Waste Disposal in the Ocean", in "Water Quality and
its Control", Vol. 5 of Hydraulic Structures Design Manual, M. Hino (Ed.), Balkema,
Rotterdam
Rutherford, J.C., 1994, "River Mixing", John Wiley & Sons, Chichester
Taylor, G.I., 1921, "Diffusion by Continuous Movements", Proc., London Math. Soc., Ser.
A, Vol. 20, pp 196-211
Taylor, G.I., 1953, "Dispersion of Soluble Matter in Solvent Flowing Slowly Through a
Tube", Proc. Royal Soc. of London, Ser. A, No. 219, pp 186-203
Taylor, G.I., 1954, "The Dispersion of Matter in Turbulent Flow Through a Pipe", Proc.
Royal Soc. of London, Ser. A, Vol. 223, pp 446-468
Yotsukura, N. and Sayre, W.W., 1976, "Transverse Mixing in Natural Channels", Water
Resources Research, Vol. 12, No. 4, pp 695-704