1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 9:
RELATIONS FOR HYDRAULIC RESISTANCE IN RIVERS
Sediment transport often creates bedforms such
as dunes. These bedforms are accompanied by
form drag, and so reduce the ability of the flow to
transport sediment.
Dunes in the Mississippi River, New
Orleans, USA
Image from LUMCON web page:
http://weather.lumcon.edu/weatherdata/audubon/map.html
Dunes on an exposed point bar in
the meandering Fly River, Papua
New Guinea
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SKIN FRICTION AND FORM DRAG: THE CONCEPTS
The drag force acting on a body can be decomposed into skin friction and form
drag. The former is generated by the viscous shear stress acting tangentially to
the body. The latter is generated by the normal stress (mostly pressure) acting on
a body. The Newtonian constitutive relation for water is
ij  pij   v,ij
,
 v,ij
 ui u j 

 

 x x 
i 
 j
Here ij denotes the stress acting in the jth direction on a face normal to the ith
direction, p denotes the pressure, ij denotes the Kronecker delta ( = 1 if i = j and 0
if i  j), ui = (u1, u2, u3) denotes the velocity vector and xi = (x1, x2, x3) denotes the
position vector. The drag force Di on a body is given as
Di    jin jdS
where ji is evaluated at the surface of the body, ni denotes a local unit vector
outward normal to the surface of the body, and dS denotes an infinitesimal element
of surface area.
2
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SKIN FRICTION AND FORM DRAG: THE CONCEPTS contd.
The drag force Di can be decomposed into a component due to skin friction Dsi and
a component due to form drag Dfi as follows:
Di  Dsi  Df i
 ui u j 
n jdS , D f i   pnidS
Dsi    v, jin jdS    


 x x 
j
i


Drag due to skin friction consists of that part of the drag that pulls the surface of the
body tangentially. Form drag consists of that part of the drag that pushes the body
in normally. Only the former is thought
to directly
contribute to sediment transport.


Now in the diagrams below let Ds and D f denote the skin friction and form drag
forces on the area element dS, n̂tx denote a unit tangential vector to the surface in
the x direction and n̂n denote a unit vector normal to the surface.
p

u
dDs  
n̂ tx dS
z body
u
z
x
n̂tx
dS

dDf  p body n̂ndS
n̂n
dS
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SKIN FRICTION AND FORM DRAG: THE CONCEPTS contd.
Let D denote the drag force in the flow direction and nx denote the component of the
unit outward normal vector to the surface in the flow direction. At sufficiently high
Reynolds number, the drag on a streamlined body is mostly skin friction. The drag
on a blunt body behind which flow separation occurs is mostly form drag. (The
pressure in the separation bubble equilibrates with the low pressure at the point of
separation.)
separation
bubble
p
u
z
flow
flow
p
p
u
D  Ds   
u
dS
z body
D  Df   p body nx dS
4
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EINSTEIN DECOMPOSITION
Einstein (1950); Einstein and Barbarossa (1952)
When bedforms are not present, all of the drag on the bed is skin friction. This
tangential drag force acts to pull the sediment along. When bedforms such as
dunes are present, part of the drag is form drag associated with (most prominently)
flow separation behind the dunes. Since this form drag is composed of stress that
acts normal to the bed surface, it does not contribute directly to the motion of bed
grains. As a result it is usually subtracted out in performing bedload calculations.
H
U
separation bubble
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EINSTEIN DECOMPOSITION contd.
Consider an equilibrium (normal) flow over a bed with mean streamwise slope S
that is covered with bedforms. The flow has average depth H and velocity U
averaged over depth and the bedforms. The boundary shear stress averaged
over the bedforms is given by the normal flow relation
b  C f U2  gHS
H
U
6
separation bubble
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EINSTEIN DECOMPOSITION contd.
Now smooth out the bedforms, “glue” the sediment to the bed so it remains flat but
offers the same microscopic roughness as the case with bedforms, and run a flow
over it with the same mean velocity U and bed slope S. In the absence of the
bedforms, the resistance is skin friction only. Due to the absence of bedforms the
skin friction coefficient Cfs and the flow depth Hs should be less than the
corresponding values with bedforms.
bs  C f sU2  gHsS
Skin friction + form drag
H
U
Skin friction only
Hs
U
separation bubble
The difference between the two characterizes form drag.
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EINSTEIN DECOMPOSITION contd.
bf = b - bs = mean bed shear stress due to form drag of bedforms
Cff = Cf – Cfs = friction coefficient associated with form drag
Hf = H – Hs = mean depth associated with form drag
bs  C f sU2  gHsS
bf  Cf fU2  gHf S
bs  bf  C f s  C f f U2  gHs  Hf S
b  C f U2  gHS
Skin friction + form drag
H
U
Skin friction only
Hs
U
separation bubble
The difference between the two characterizes form drag.
8
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SKIN FRICTION
Skin friction can be computed using the techniques developed in Chapter 5;
where  = 0.4 and r = 8.1,
C
1/ 2
fs
1  Hs 
 n11 
  ks 
1/ 6
C
or
k s  nk Ds90
1 / 2
fs
 Hs 
  r  
 ks 
, nk  2
bs  C f sU2  gHsS
Skin friction + form drag
H
U
Skin friction only
Hs
U
separation bubble
The difference between the two characterizes form drag.
9
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORM DRAG OF DUNES: EINSTEIN AND BARBAROSSA (1952)
One of the first relations developed to predict the form drag in rivers in which dunes
predominate is that of Einstein and Barbarossa (1952). They obtained an empirical
form for Cff as a function of s*, where
bs
u2s


RgD 35 RgD 35
denotes the Shields
number due to skin
friction and D35 is the
grain size such that 35
percent of a bed surface
sample is finer. Note
that
u s 
bs

0.1
0.01
Cff


s 35
0.001
0.0001
0.01
0.1
1
s35*
10
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORM DRAG OF DUNES: ENGELUND AND HANSEN (1967)
The total shear velocity u*, shear velocity due to skin friction u*s and shear velocity
due to bedforms u*f, and the associated Shields numbers are defined as
u 
b

, us 
bs

bf

, u f 
u2
 
RgD s50

u2s
 
RgD s50

s
,
,
u2f
 
RgD s50

f
Engelund and Hansen (1967) determined the following empirical relation for lowerregime form drag due to dune resistance;

s
 
  0.06  0.4 
or thus

f


s


 2
 
 2
        0.06  0.4 
Note that bedforms are absent (skin friction only) when s* = *; bedforms are
present when s* < *. The relation is designed to be used with the following
skin friction predictor:
1  Hs 
k s  2 Ds65
n11 
  ks 
Engelund and Hansen (1967) also present a form drag relation for upperregime bedforms (antidunes).
Cf s1/ 2 
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORM DRAG OF DUNES: ENGELUND AND HANSEN (1967) contd.
Engelund-Hansen Bedform Resistance Predictor
3
2.5
No form drag
2
x
s
E-H Relation
No form drag
1.5
f
1
0.5
Engelund-Hansen
s
0
0
0.5
1
1.5

x

2
2.5
3
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DEPTH-DISCHARGE PREDICTIONS WITH THE FORM DRAG PREDICTOR OF
ENGELUND AND HANSEN (1967)
Form drag relations allow for a prediction of flow depth H and velocity U as a
function of water discharge per unit width qw. In order to do this with the relation of
Engelund and Hansen (1967) it is necessary to specify the stream slope S, bed
material sizes Ds50 and Ds65, submerged specific gravity of the sediment R. The
computation proceeds as follows for the case of normal flow, for which b = u*2 =
gHS.
Compute ks from Ds65.
Assume a value (a series of values) of Hs.
Assuming normal flow, compute u*s = (gHsS)1/2 and s* =u*s2/(RgDs50).
Compute * from s* according to Engelund-Hansen.
Again assuming normal flow, * = (HS)/(RDs50) so that H = RDs50*/S.
Compute Czs = Cfs-1/2 from Hs/ks and the skin friction predictor.
Compute the velocity U from the relation U/u*s = Czs.
Compute the water discharge per unit width qw = UH.
Plot H versus qw.
13
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORM DRAG OF DUNES: WRIGHT AND PARKER (2004)
The form drag predictor of Engelund and Hansen (1967) tends to work well for
sand-bed streams at laboratory scale. It also works well at small to medium field
scale, i.e. in streams in which dunes give way to upper-regime plane bed before
bankfull flow is achieved. It works rather poorly for large, low-slope sand-bed
rivers, in which dunes are usually never washed out even at or above bankfull
flow. Wright and Parker (2004) have modified it to accurately cover the entire
range.


s

  0.05  0.7  Fr

0.7 0.8
U
Fr 
 Froude number
gH
This relation is designed to be used with the skin friction predictor
C
1/ 2
fs
8.32  Hs 
 

 strat  k s 
1
6
k s  3Ds90
where strat is a correction for flow stratification which can be set equal to unity in
the absence of other information (see original reference).
14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COMPARISON OF FORM DRAG PREDICTORS AGAINST FIELD DATA
The Niobrara and Middle Loup are small sand-bed streams. The Rio Grande is a
middle-sized sand-bed stream. The Red, Atchafalaya and Mississippi Rivers are
large sand-bed streams.
2.0
Middle Loup
Rio Grande
Atchafalaya
Engelund-Hansen
1.8
1.6
2.0
Niobrara
Red
Mississippi
1.6
1.4

 * sk
s

Middle Loup
Rio Grande
Atchafalaya
New relation
1.8
Niobrara
Red
Mississippi
1.4
1.2

 * sks

 * sk   *
1.0
0.8
1.2
1.0
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.5
1.0
1.5
 *
2.0
2.5

Engelund and Hansen (1967)
3.0
0.0
0.0
0.5
1.0
1.5
 * Fr 0.7

 Fr
2.0
2.5
0 .7
Wright and Parker (2004)
15
3.0
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DEPTH-DISCHARGE PREDICTIONS WITH THE FORM DRAG PREDICTOR OF
WRIGHT AND PARKER (2004)
The relations can be written
as:
or alternatively as:
H 
U
 8.32  s 
gHsS
 ks 
H 
U  8.32 gHsS  s 
 ks 
1
6
1
6
 HS   U 0.7 
 
 
s  0.05  0.7 

 RD s50   gH  


 RD
s 50
H  
S


 g


 U 


0 .7




20 / 13
 s  0.05 

  
0
.
7


5/4
The computation proceeds as follows for the case of normal flow, for which b =
u*2 = gHS. The stratification correction is not implemented here for simplicity.
Compute ks from Ds90.
Assume a value (a series of values) of Hs.
Assuming normal flow, compute u*s = (gHsS)1/2 and s* =u*s2/(RgDs50).
Compute the velocity U from the skin friction predictor.
Compute  from the indicated equation.
Compute H from the indicated equation.
Compute the water discharge per unit width qw = UH.
Plot H versus qw.
0 .8
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PREDICTION OF BEDLOAD TRANSPORT IN A STREAM IN WHICH DUNES
MAY BE PRESENT
If dunes are not present, the calculation of bedload transport may proceed using the
techniques of Chapter 7.
If dunes are present, the calculation is based not on the total boundary shear stress
b, but rather just that component due to skin friction bs. Thus in the case of
relations for uniform sediment D, the following transformation must be made
bs
u2s
  

RgD RgD


s
so that the bedload relation of e.g. Ashida and Michiue (1972) is recast as

qb  17 s  c


s  c , c  0.05
In the case of the normal flow
HsD
 
RD

s
and the calculation can proceed from the calculation of the
depth-discharge relation.
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE PREDICTION OF FLOW AND BEDLOAD TRANSPORT
Depth-Discharge and Bedload Calculator
Input Parameters
S
4.00E-05
D50
0.3 mm
D90
0.6 mm
nk
3
R
1.65
s 
HsS
RD 50
H
U  8.32 gHsS  s
 ks
Uses
a) Wright-Parker formulation for flow resistance (without stratification correction)
b) Ashida-Michiue formulation for bedload transport,
bed slope
median sediment size
size such that 90% of the sediment is finer
factor such that ks = nk Ds90
submerged specific gravity of sediment



1
6
 RD
s 50
H  
S


qw
   0.05 

   s
0 .7


5/4
 g


 U 


 UH
0 .7




20 / 13
 
Fr 
HS
RD50
This calculation is implemented in:
Rte-bookWPHydResAMBL.xls
U
gH
u 
qb  RgD50 D50 ( s  0.05)( s  0.05 )
gHS
us 
gHsS
Discard first
three rows
Input
Hs (m) s*
U (m/s) 
H (m) qw (m2/s)*
s*/* Fr
u* (m/s) u*s (m/s) qb (m2/s)
0.800 0.0646
0.4072 0.008 0.254 0.1036 0.0206 3.145 0.258 0.00999 0.01772 1.60E-07 Discard results whenever s*/* > 1
0.900 0.0727
0.4405 0.014 0.544 0.2397 0.0440 1.654 0.191 0.01461 0.01879 3.72E-07
1.000 0.0808
0.4725
0.02 0.906 0.4279 0.0732 1.104 0.159 0.01885 0.01981 6.64E-07
1.100 0.0889
0.5035 0.027 1.323 0.6664 0.1069 0.831
0.14 0.02279 0.02078 1.03E-06
1.200 0.0970
0.5336 0.034 1.788 0.9539 0.1444 0.671 0.127 0.02648 0.0217 1.47E-06
1.300 0.1051
0.5629 0.042
2.29 1.2891 0.1851 0.568 0.119 0.02998 0.02259 1.97E-06
1.400 0.1131
0.5914 0.049 2.826 1.6712 0.2284 0.495 0.112 0.0333 0.02344 2.53E-06
1.500 0.1212
0.6192 0.057
3.39 2.0994 0.2740 0.442 0.107 0.03647 0.02426 3.15E-06
1.600 0.1293
0.6464 0.066
3.98 2.5729 0.3216 0.402 0.103 0.03952 0.02506 3.83E-06
1.700 0.1374
0.6731 0.074 4.592 3.0912 0.3711
0.37
0.1 0.04245 0.02583 4.57E-06
1.800 0.1455
0.6992 0.083 5.225 3.6536 0.4222 0.344 0.098 0.04528 0.02658 5.35E-06
1.900 0.1535
0.7249 0.092 5.876 4.2598 0.4749 0.323 0.095 0.04802 0.0273 6.19E-06
2.000 0.1616
0.7501 0.101 6.545 4.9092 0.5289 0.306 0.094 0.05068 0.02801 7.08E-06
2.100 0.1697
0.7749
0.11 7.229 5.6015 0.5841 0.291 0.092 0.05326 0.02871 8.01E-06
2.200 0.1778
0.7993 0.119 7.927 6.3362 0.6406 0.278 0.091 0.05577 0.02938 8.99E-06
2.300 0.1859
0.8234 0.129 8.639 7.1130 0.6981 0.266 0.089 0.05822 0.03004 1.00E-05
2.400 0.1939
0.8471 0.138 9.364 7.9316 0.7567 0.256 0.088 0.06062 0.03069 1.11E-05
18
2.500 0.2020
0.8704 0.148
10.1 8.7917 0.8162 0.248 0.087 0.06296 0.03132 1.22E-05
The basis for the
calculation is a
large sand-bed
stream. The
calculation uses
Wright-Parker
(without
stratification
correction) and
Ashida-Michiue.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DEPTH-DISCHARGE AND BEDLOAD RELATION FOR SAMPLE CALCULATION
Wright-Parker depth-discharge predictor: Ashida-Michiue bedload transport relation
10.0
1.4E-05
9.0
1.2E-05
8.0
1.0E-05
2
H (m), qb (m /s)
7.0
6.0
8.0E-06
5.0
6.0E-06
4.0
3.0
H
qb
4.0E-06
2.0
2.0E-06
1.0
0.0
0.0
2.0
4.0
6.0
qw (m2/s)
8.0
0.0E+00
10.0
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A BULK PREDICTOR FOR DEPTH-DISCHARGE RELATIONS
The Brownlie (1982) empirical depth-discharge predictor has been demonstrated to
be accurate for both laboratory and field sand-bed streams. It takes the lowerregime form
1
0.6539
0.09188
0.1050
Ĥ  0.3724 S (q̂w S)
S
g
and the upper-regime form
where
Ĥ  0.2836 S1 (q̂w S)0.6248 S0.08750 0g.08013
Ĥ 
H
Ds50
, q̂w 
qw
gDs50 Ds50
Once H is known U = qw/H can be computed. It is then possible to back-calculate
Hs from any appropriate relation for skin friction and the normal flow assumption,
e.g.
1/ 6
 Hs 
U
 r  
gHsS
 ks 
Once Hs is known, s* = (HsS)/(RDs50) and thus the bedload transport rate can be
computed. A discriminator between lower regime and upper regime can
20
be found in the original reference.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ANOTHER BULK PREDICTOR FOR FLOW RESISTANCE
The flow predictor of Karim and Kennedy (1981) takes the following form:

U
og10 
 RgD
50




q
t
  0.9045  0.1665 og10 


 RgD D 
50
50 




 u 
q
 uD50 
t

 0.0831og10   og10 
 og10 
 RgD D 
  
 vs 
50
50 

 u 
u D 
 0.2166 og10   og10   50 
  
 vs 
 H 
u 
 og10 S  103 og10   
 0.0411og10 
 D50 
 vs 


where qt denotes the total volume bed material load per unit width. Karim and
Kennedy’s predictor for qt is presented in Chapter 11.
21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GENERALIZATION TO GRADUALLY VARIED FLOWS
The preceding calculations are predicated on the assumption of normal flow. In the
case of gradually varied flow, the equation to be solved is
dH S  S f

dx 1  Fr 2
In the calculation of gradually varied flow the actual slope S should be replaced by
the friction slope Sf in the relations for skin friction and form drag:

U2
S f  Cf
 b
 b  gSf H
gH gH
For example, the relations of Wright and Parker (without stratification correction)
become
 Hs 
qw
 8.32  
H gHsS f
 ks 
1
6
0.7





HsS f
HS
q
f
w
 
 
 0.05  0.7 
3
/
2
 RD s50   g H  
RD 50

 

0 .8
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GENERALIZATION TO GRADUALLY VARIED FLOWS contd.
The flow is assumed to be subcritical. The depth H is assumed to be known at the
downstream point H2; it is to be computed at the upstream point H1. The
formulation can be discretized as
 




1
2
1
2
 Sf1
 Sf 2 

1  x
x

H1  H2  x

2
2
qw
qw
2 

1

1

3
3


gH
gH
1
2


flow
Now since qw and H2 are known, Hs1 and
Sf1 can be computed (iteratively) from the
two relations
H1
H2
x
x1
x2
H 
qw
 8.32  s1 
H1 gHs1S f 1
 ks 
1
6
0.7
 H S   q


Hs1S f 1
1
f
1
w
 
 
 0.05  0.7 
3/2 

RD 50
 RD s50   g H1  


0.8
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GENERALIZATION TO GRADUALLY VARIED FLOWS contd.
Once all quantities at x2 are computed, H1, Hs1 and Sf1 can be computed iteratively
from the following three equations.
 

1  2
1
2
 Sf1
 Sf 2 
1  x

H1  H2  x
 x 2
2
qw
qw
2 

1

1


gH13
gH32 

flow
 Hs1 
qw

 8.32 
H1 gHs1S f 1
 ks 
H1
1
6
H2
0.7






Hs1S f 1
H
S
q
1
f
1
w
 
 
 0.05  0.7 
3/2 

RD 50
 RD s50   g H1  


x
0.8
24
x1
x2
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 9
Ashida, K. and M. Michiue, 1972, Study on hydraulic resistance and bedload transport rate in
alluvial streams, Transactions, Japan Society of Civil Engineering, 206: 59-69 (in Japanese).
Brownlie, W. R., 1981, Prediction of flow depth and sediment discharge in open channels, Report
No. KH-R-43A, W. M. Keck Laboratory of Hydraulics and Water Resources, California
Institute of Technology, Pasadena, California, USA, 232 p.
Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel
Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service.
Einstein, H. A and Barbarossa, N. L., 1952, River Channel Roughness, Journal of Hydraulic
Engineering, 117.
Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams,
Technisk Vorlag, Copenhagen, Denmark.
Karim, F., and J. F. Kennedy, 1981, Computer-based predictors for sediment discharge and
friction factor of alluvial streams, Report No. 242, Iowa Institute of Hydraulic Research,
University of Iowa, Iowa City, Iowa.
Wright, S. and Parker, G., 2004, Flow resistance and suspended load in sand-bed rivers:
simplified stratification model, Journal of Hydraulic Engineering, 130(8), 796-805.
25