Appendix C. SYNTHETIC CHANNEL SPREADSHEET
C.1
OVERVIEW
The Synthetic Channel EXCEL Spreadsheet simulates the hydraulic behavior of an ideal
channel cross section. The user specifies the channel shape, bankfull dimensions, slope,
and bed-material size, and then can examine characteristics of flows within that channel
by specifying a range of central (maximum) flow depths, which are equivalent to watersurface elevations or stages. The basic model is on the worksheet labeled “SynChan” and
the model output can be assembled for tabular or graphical presentation on the worksheet
labeled “GraphData”.
The model can be used to explore the general nature of important hydraulic
relations and ways in which these relations change with channel shape, dimensions,
slope, and bed-material size, including:
1.
At-a-station hydraulic-geometry relations
2.
Flow resistance − discharge relations
3.
Discharge (or depth) at which erosion begins
4.
Stage-discharge (rating-curve) relations
5.
Froude-number−discharge relations
6.
Reynolds-number−discharge relations
7.
Cross-channel distribution of surface velocity
8.
Distribution of velocity throughout the flow
9.
Energy and momentum coefficients
10.
Ratio of mean to maximum velocity
2
11.
Effects on hydraulic characteristics of assuming various vertical-velocity profiles
12.
Effects of channel shape on hydraulic relations
13.
The effects of water temperature on hydraulic relations
C.2
BASIC APPROACH
C.2.1 Channel Shape
The channel cross-section is symmetrical with its shape determined by the user-specified
value of the exponent r 1in the general cross-section model described in section 2.4.3.2
(equation 2.20):
r
z(w) = Ψ BF
 2w 
 , 0  w  WBF /2,
 
W
 BF 
(C.1)
where z(w) is the elevation of the channel bottom at cross-channel distance w from the
center, ΨBF is the user-specified maximum (central) bankfull depth, and WBF is the userspecified bankfull width. For a triangular channel, r = 1; for the Lane stable channel, r =
1.75; for a parabolic channel, r = 2; and the channel shape approaches a rectangle as r →
∞. (A rectangle can be approximated by using a large value for r, say r = 10,000.) Values
of r < 1 (“convex channels”) can also be specified.
In the model, rectangular elements of one-half of the symmetrical cross section
are represented by spreadsheet cells. The width of each element is equal to WBF /200 and
the height is equal to ΨBF /100.
1
User-specified quantities are shown in bold herein.
3
C.2.2 Velocity
Each cell that is below the water surface and above the channel bottom displays the local
velocity; other cells are blank. In the default version of the model, the local velocities
uw(y) are computed by the Prandtl-von Kármán (P-vK) velocity profile for turbulent flow
(equation 5.21),
1
uw(y) =    ( g  Yw  S s )1 / 2 ∙ln
 
 y 

 ,
y
 0w 
(C.2)
where y is distance above the channel bed, κ is von Kármán’s constant (κ = 0.4), g is
gravitational acceleration (g = 9.81 m/s2), Yw is the local water depth, Ss is the userspecified water-surface slope. As described in section 5.3.1.6 (equation 5.32), the value
of y0w is determined by the value of the local boundary Reynolds number, Rebw,
Rebw 
u* w  y r


( g  Yw  S 0 )1/2  y r

,
(C.3)
where u*w is the local friction velocity, yr is the effective height of bed roughness
elements, and ν is kinematic viscosity:
if Reb ≤ 5 (smooth flow), y0w =

9  u *w
;
if Rebw > 5 (transitional or rough flow), y0w =
(C.4a)
yr
,
30
(C.4b)
4
and yr is considered equal to the user-specified 84th-percentile of the bed-material grain
size, d84 (equation 6.2).
Note that it is a simple matter to replace the Prandtl-von Kármán (P-vK) profile
by one of the other profiles discussed in sections 5.3.2 − 5.3.5, as described below in
section C.3.2.
C.2.3 Water Properties
The values of water properties mass density, ρ; weight density, γ; dynamic viscosity, μ;
and kinematic viscosity, ν, are required to compute some of the flow characteristics.
These properties are functions of the user-specified water temperature, T, and are
computed via equations 3.11 and 3.20.
C.3
MODEL OPERATION AND DISPLAYS
C.3.1 User-Specified Quantities
The user enters the quantities describing a particular channel in the cells colored light
yellow. These quantities are
Quantity
r
ΨBF
WBF
S0
d84
Units
[1]
m
m
[1]
mm
Cell
F1
B3
B4
E3
E4
The stage increments are numbered from 0 at the bed to 100 at ΨBF in cells A30:A130,
and the corresponding stage values (Z = Ψ − z, m) are in cells B30:B130. The channelwidth increments are numbered from 0 at the center to 100 at WBF /2 in cells D25:CZ25,
5
and the corresponding distances from the center (w, m) are in cells D26:CZ26. The
elevation of the channel bottom [z(w), m] at each width increment is computed via
equation 2.20 and displayed in cells D27:CZ27.
The value of the water temperature T is entered in cell I5 and the water properties
at T are displayed in cells J5:M5; densities are computed via equation 3.11 and
viscosities via equation 3.20.
Once the channel characteristics and water temperature are specified, the model
calculates the hydraulic properties of a particular flow within that channel corresponding
to a particular value of the maximum water depth Ψ, which the user specifies in cell B6
(colored yellow) (0 ≤ Ψ ≤ ΨBF).
C.3.2 Elemental Velocities
The velocity uw(y) (m/s) within each cross-section element that lies between an elevation
equal to the channel bed plus the d84 value and the water surface are calculated and
displayed within the field of cells D30:CZ130. In the default version of the model, the
Prandtl-von Kármán velocity profile [equation (5.21)] is used and the default formula for
calculating velocity is written in cell D30 as
=IF($C30<=$B30,IF($C30>(D$27+$E$4/1000),
2.5*SQRT(9.81*$E$3*(MAX($C$30:$C$130)-D$27))*LN(($C30-D$27)/D$24)," ")," ")
and copied into cells D30:CZ130. The underlined portion of the above code is the
Prandtl-von Kármán formula. Alternative profiles, such as described in sections 5.32 −
6
5.34, can be used in the Synthetic Channel by replacing the underlined code
appropriately. For example, the hyperbolic tangent profile of equation 5.47 can be
entered as
IF($C30<=$B30,IF($C30>D$27,
4.5*SQRT(9.81*$E$3*MAX(($C$30:$C$130)−D$27))*
TANH(1+(($C30−($E$4/1000))/($E$4/1000)))," ")," ")
C.3.3 Hydraulic Quantities at Each Width Increment
A number of hydraulic quantities averaged or summed for the number of cells with flow
( N) at each width increment w are calculated and displayed in cells D15:CX24:
Symbol
ΔPw
Qw
Aw
Yw
Cells avg Uw
P-vK avg Uw
τ0w
u*w
Rebw
y0w
Quantity
Wetted-perimeter increment (m)
Discharge (m3/s)
Flow area (m2)
Flow depth (m)
Average velocity (m/s)
Average velocity (m/s)
Boundary shear stress (N/m2)
Friction velocity (m/s)
Boundary Reynolds number
P-vK fitting parameter
Calculated as
[(wi − wi+1)2 + (ΨBF /100)2] 1/2
(ΨBF /100)∙(WBF /200)∙uw(y)
(ΨBF /100)∙(WBF /200)
Ψ − z(w)
uw(y)/N
equation 5.40
γYw∙S0
(g∙Yw∙S0)1/2
equation 5.31
equation 5.32
C.3.4 Flow Hydraulic Properties
C.3.4.1 Erosion?
The model calculates the fall velocity vf for the specified d84 sediment size via equation
12.21 and displays it in cell E5. It then applies the Rouse-number erosion criterion
described in section 12.5.2.1 to determine if erosion is occurring at the channel center (w
7
= 0), where the friction velocity (or shear stress) has its maximum value. According to
this criterion, if the Rouse number Ro  vf /(κ∙u*0) < 5.4, erosion is occurring and “YES”
appears in cell E6 and a warning message appears in cell E8. If erosion is occurring, the
channel geometry is changing with time and the model computations, which assume a
constant channel shape, are not applicable.
C.3.4.2 Overall Flow Quantities
The major quantities of interest characterizing the entire flow (cross-section-averaged or totaled) are displayed in cells A10:T13. The quantities displayed in row 12 are calculated
via theoretical formulas discussed in the text; for comparison, the corresponding
quantities in row 13 are determined by averaging or summing values in the N spreadsheet
cells for which velocity is calculated (empty cells are not included). The following table
gives the formulas used to compute these values:
Column→
Row↓
A
Q
B
A
 r   WBF  ( r 1) / r


  
  1/ r   
r

1

  BF 
(table 2.7)
12
13
C
W
Qw
Aw
  

W BF  
 BF 
(table 2.7)
D
Pw
1/ r
N∙WBF/200
[(wi − wi+1)2
+ (ΨBF /100)2]
1/2
Column→
Row↓
12
13
E
Y
 r 

  
 r  1
(table 2.7)
Yw/N
F
R
A/Pw
G
W/Y
H
U
W/Y
Q/A
W/Y
Uw/N
8
Column→
I
J
K
Row↓
u*
Ω
Ω*

 y
1.95  
0.400∙  ln  r  0.9  
 11  Y Re  

(eqn. 6.24)
(g∙Y∙S0)1/2
u*/U
12
L
  *
*
(eqn. 6.16b)
1
  *
*
(section 6.6)
(g∙Y∙S0)1/2
u*/U
13
(eqn. 6.16b)
Column→
Row↓
12
M
N
O
P
nM
C
Fr
Re
R 2 / 3  S 1/ 2
U
U
0.552  R 1 / 2  S 1 / 2
U
( g  Y )1 / 2
Y U
(eqn. 6.40b)
(eqn. 6B1.3)
(eqn. 6.5)
(eqn. 3.43)

13
Column→
Row↓
Q
R
S
T
umax
U/umax


1 + SK[uw(y)]∙CV3 [uw(y)]
+ 3∙ CV2[uw(y)]
1 + CV2 [uw(y)]
12
13
Maximum value
in cells D30:D130
U/umax
(eqn. 8B2.14)
(eqn. 8B2.15)
C.3.5 Tables and Graphs
Tables and graphs relating hydraulic parameters can be constructed from the worksheet
labeled “GraphData”. On this worksheet, the selected value of Ψ is displayed in cell A1
and the corresponding flow properties computed in cells A12:T13 of the “SynChan”
worksheet are reproduced in cells B1:U1. To make tables or graphs showing how these
9
properties change as flow changes within a given channel, the quantities in cells A1:U1
(highlighted in light yellow) can be copied into successive rows below the labeled
headings in cells A6:U6 using the “Paste Special, Values” command. Once the data are
successively copied into these cells, tables and graphs comparing hydraulic properties of
interest can be constructed. Graphs are constructed using the “Chart Wizard, XY
(Scatter)” facility.
C.4
EXAMPLE USES
Here we show how the Synthetic Channel model can be used to explore the forms of
some important hydraulic relations (section 2.6.3.1) in a parabolic channel.
C.4.1 Enter Channel Properties and Water Temperature
The channel characteristics and water temperature are entered in the “SynChan”
worksheet as follows:
Quantity
r
ΨBF
WBF
S0
d84
T
Units
[1]
m
m
[1]
mm
o
C
Cell
F1
B3
B4
E3
E4
I5
Value
2
1.00
20.00
0.001
10
10
C.4.2 Enter Stage Values
A value of maximum depth Ψ = 0.1 m is entered in cell B6, and the corresponding flow
characteristics are computed and displayed in cells A12:T13 of the “SynChan” worksheet
and A1:U1 of the “GraphData” worksheet. These values are transferred to cells A8:U8
using the “Copy” and “Paste Special, Values” commands. Then a value of Ψ = 0.2 m is
10
entered in cell B6 of “SynChan”, and the process is repeated, this time transferring the
quantities in cells A1:U1 to cells A9:U9 of “GraphData”. Successive values Ψ = 0.3, 0.4,
…, 1.0 m are then entered in cell B6 of “SynChan”, and the values in A1:U1 of
“GraphData” successively transferred to rows A10:U10, A11:U11, …, A17:U17.
Then using the Chart Wizard, we can construct the graphs of the relations of
interest, as described in the following sections.
C.4.3 At-a-Station Hydraulic Geometry
The generated data tables can be used to explore at-a-station hydraulic-geometry relations
(i.e., width, W ; average depth, Y; and average velocity, U; vs. discharge, Q; see section
2.6.3.1) for this reach, as shown in the following graphs.
100
Wy ==10.664x
10.7∙Q 0.232
0.2317
2
Width, W (m)
R = 0.9956
10
1
0.01
0.1
1
Discharge, Q (m3/s)
10
100
11
1
0.4369
0.2005x
0.437
Y =y =0.200∙Q
R = 0.9966
Depth, Y (m)
2
0.1
0.01
0.01
0.1
1
10
100
10
100
3
Discharge, Q (m /s)
10
y = 0.4799x 0.320
U = 0.480∙Q
0.3199
Velocity, U (m/s)
R2 = 0.9829
1
0.1
0.01
0.1
1
3
Discharge, Q (m /s)
12
These plots are similar to actual hydraulic-geometry relations, as shown for
example in figure 2.42. The values of the exponents in these relations, b = 0.232, f =
0.437, m = 0.320, plot very close to the “average” value for natural channels and very
close to the theoretical line for parabolic channels (r = 2) shown on the triaxial diagram
of figure 2.41. (Note that the hydraulic-geometry equations shown on the graphs are
readily generated via the “Chart, Chart Options, Add Trendline, Power” commands in
EXCEL.)
The model could then be used to explore how these relations change with channel
shape, slope, bed-material size, or other factors.
C.4.4 Resistance Relations
We can use the tabulated data to construct graphs of resistance, , and Manning’s
resistance factor, nM, vs. discharge, Q:
0.09
0.08
0.07
 vs. Q
Resistance
0.06
0.05
0.04
0.03
n M vs. Q
0.02
0.01
0
0
2
4
6
8
10
Discharge (m3/s)
12
14
16
18
20
13
The Y values displayed on the “GraphData” worksheet can be divided by the bedmaterial size to explore the relation of  and nM to Y/d84:
0.09
0.08
0.07
 vs. Y /d 84
 and nM
0.06
0.05
0.04
0.03
n M vs. Y /d 84
0.02
0.01
0
0
10
20
30
40
50
60
70
Y/d84
These relations are very similar to those observed in natural channels, and to figure 6.9.
C.4.5 Froude Number − Discharge Relations
The Synthetic Channel can be used to show how the Froude number, Fr, typically varies
with discharge:
14
0.7
0.6
Froude No.
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
3
Discharge (m /s)
This pattern is very typical of Froude-number − discharge relations seen in natural
channels.
C.5
CAVEAT
Although the hydraulic relations computed by the Synthetic Channel model are similar in
form to corresponding relations in natural channels,
THE MODEL DOES NOT SIMULATE THE EXACT QUANTITATIVE RELATIONS
OF ACTUAL CHANNELS AND SHOULD NOT BE USED TO PREDICT THOSE
RELATIONS.
15
The principal reason for this is that the model captures only the effect of
resistance originating on the channel bed and transmitted throughout the flow via the
mechanisms modeled by the Prandtl-von Karman velocity profile (section 5.3.1) (or
alternative profile). Effects on resistance of other factors, such as irregularities in crosssection shape upstream, downstream, and in the cross section), longitudinal profile,
planform, vegetation, surface instabilities, sediment, bedforms, etc. (see section 6.6),
which are always present in real channels, are not simulated in the model. This is
reflected in the values of Manning’s nM calculated for the synthetic channel, which are
usually are considerably lower than observed in natural channels (see table 6.5).