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1 Senior Engineer, Mussetter Engineering Inc., Fort Collins, CO.
2 Professor, Dept. of Civ. Engrg. Colorado State University, Fort Collins, CO.
CFD for Predicting Shear Stress in a Curved Trapezoidal ChannelBy Daniel Gessler1, Robert N. Meroney2
Abstract
Continuing advances in computer speed and the availability of user friendly software has
made computational fluid dynamics a cost effective compliment and at times alternative to physical
modeling in the field of civil engineering hydraulics. Validation of the CFD models for open
channel flow conditions remains limited due to the significant cost of obtaining data.
Furthermore, once the flume study has been conducted, a numerical study is typically no longer
required.
This paper explores the ability of CFD to reproduce free surface flow in a trapezoidal
channel in a bend. Laboratory data was collected at MIT during the late 1950's and early 1960's to
obtain an understanding of shear stress distribution and flow patterns in bends. A portion of those
studies was reproduced using CFD and a comparison made between observed and predicted values.
The objective of the paper is to demonstrate that CFD can offer a cost effective alternative and
compliment to physical modeling of smooth, rigid boundary conditions.
The CFD software Fluent (1998) was used to model flow in a trapezoidal channel using two
turbulence models, K-Epsilon and Reynolds Stress. Model results show that observed and predicted
water surface elevations typically differ by less than 2.5 percent, and predicted shear stresses differ
by less than 10 percent. Model results are also used to illustrate the limitations of the K-Epsilon
model as well as conditions under which it produces very similar results to the significantly more
expense Reynolds Stress model. Results of the study clearly demonstrate the strengths and
weaknesses of using CFD as an alternative or compliment to physical modeling.
Introduction
Advances in computer speed and user friendly software has made the use of computational
fluid dynamics (CFD) viable for engineers and researchers. Complex fluids problems can be solved
using a desk side computer and commercially available software. The use of CFD is wide spread
in mechanical and aerospace engineering. However, in the field of civil engineering and particularly
in hydraulics, there is minimal utilization of CFD. A lack of validation and test cases appears to
make hydraulic engineers reluctant to accept modeling results.
The objective of this paper is to explore and demonstrate the strength of CFD modeling on
an open channel flow problem. A data set collected during the late 1950's at the Massachusetts
Institute of Technology is used as a basis for model validation. Ippen et al.(1960, 1962a, and 1962b)
published the results of an investigation to quantify the boundary shear stress distribution in curved
trapezoidal channel. Flow through a single curve and a simulated compound curve in a smooth
boundary trapezoidal channel was studied. Boundary shear stress data are presented in
dimensionless form by dividing the local shear stress, τ, by the shear stress for uniform flow, .τ o
In addition to contour maps of shear stress, velocity contours and water surface elevations are
presented. Two of the flow configurations tested by Ippen et al. (1962b) were reproduced
numerically. A comparison is made between the model results and those obtained Ippen et
al.(1962a).
Ippen et al. (1962b) developed some general guidelines on the variation of shear stress in
bends as a function of channel geometry and hydraulic parameters, however, he also indicated that
�Within a curved reach, however, the local shear stresses vary in a manner that cannot be predicted
at present (1962) because of the effects of local accelerations and of secondary motion in the flow.�
The complexity of predicting the magnitude of shear maxima in a bend over a range of flows
remains unchanged. A comprehensive set of empirical coefficients for predicting shear stress
distribution in curved channels is not known to exist. Therefore, in order to accurately determine
shear stress maxima in bends, sight specific investigations remain necessary.
Background
During the period from March 1958 to September 1961 an investigation of boundary shear
stress in curved trapezoidal channels was under taken at the Hydrodynamics Laboratory of the
Massachusetts Institute of Technology. The work was funded by the Soil and Water Conservation
Research Division, Agriculture Research Service, U.S. Department of Agriculture. The objective
of the study was to determine the magnitude and distribution of boundary shear stress in curved
reaches of prismatic channels. The work is described by Ippen et al. (1962a) , as being �... an initial
attack on the erosion problem, all sediment properties were excluded from consideration, and only
the effects of the flow pattern on a clearly defined set of boundary surfaces were studied. A general
explanation of the various related aspects of sediment mechanics - erosion, transport, and deposition
- will require much supplementary investigation of systems containing sediment ...�
Understanding the shear stress distribution in fixed bed, trapezoidal channels has little direct
application to natural stream systems. However, it provides a basis where under a set of controlled
circumstances our fundamental understanding of flow in a curved channel can be tested and
improved.
Two trapezoidal channels were used by Ippen et al. (1962b), results from one of the channels
were used for this investigation. The quality of the experimental setup is crucial as potential sources
of experimental error. The trapezoidal channel used consisted of a straight 6.096 m (20 ft) long
section with a single curve of 60 degrees central angle followed by a straight 3.048 m (10 ft) long
exit section. Figure 1, reproduced from Ippen et al. (1960), shows the trapezoidal channel. The
radius of curvature in the bend is 1.524 m (60 inches).
The side slopes of the channel were constant at 2 horizontal and 1 vertical. During the
construction of the flume, the bed slope was set at 0.000 64 = 1/1563. The slope was not adjusted
during the runs, however, it was found that some settling of the channel had caused the slope to
change to 0.000 55, a change of approximately 14 percent. The runs of interest for this investigation
were made before the settling was observed (Ippen et al., 1962b).
The channel was composed of short sections of reinforced precast concrete and supported
on a steel frame work. The channel surfaces within the concrete trough were created using plaster
and a trapezoidal scraper mounted on guide rails. The plaster was sanded and then plastic coated
to give a hard smooth finish.
A number of desired depth conditions were selected by Ippen et al. (1962b). For each depth,
the corresponding discharge was determined using the Manning the equation. Manning�s n for the
channel was initially estimated at 0.009, and later checked to be 0.010. A sluice gate at the
downstream end was used to obtained the desired water level at the upstream entrance to the curve
for a given discharge.
Shear stress measurements were made using a Pitot tube in a manner developed by Preston
(1954). Preston demonstrated that shear stress on a smooth surface could be computed from the
dynamic pressure measured by a round Pitot tube resting on the surface. Given that the tube has a
sufficiently large diameter, the effects of the very thin viscous sublayer are insignificant. The total
pressure registered by the Pitot tube is only dependent on the velocity distribution in the turbulent
boundary layer.
Ippen et al (1962b), used the following common expression for the velocity distribution in
a turbulent boundary layer over a smooth surface:
(1)uu
fu y
*
*=
υ
Where the shear velocity, , is expressed as u*
(2)u o* =
τρ
and y is the distance normal to the boundary surface, is the kinematic viscosity of the fluid, υ ρ
is the mass density of the fluid, and is the local boundary shear stress. The notation being usedτ o
is the same as that used by Ippen et al (1962b). Preston developed a functional relationship for the
boundary shear stress and directly calibrated it using pipe flow giving the follow equation:
(3)( )
log . . logτρυ ρυo t od P P d2
2
2
241396 0875
4= − +
−
The equation developed by Preston (1954) is valid in the following range:
(4)4 54
652
2. log( )
.⟨−
⟨P P dt o
ρυ
Where is the dynamic pressure recorded by a round Pitot tube of diameter, d. The( )P Pt o−
calibration equation developed by Preston is valid only when the velocity distribution near the wall
follows the following dimensionless form:
(5)uu
u y*
*.=
8 61
17
υ
Ippen et al. (1962b), directly calibrated the surface Pitot they used in a tilting flume. The
calibration was valid only in straight flumes, since it was not possible to directly calibrate for the
curved sections. Therefore, Ippen et al. (1962b) carefully measured the velocity profile at 1 mm
intervals at various points in the curved sections of the flume with a flat tipped Pitot tube and
compared the distributions to equation 5. Finally, in preliminary testing, the sensitivity of the
instrument to a moderate misalignment with the local flow vector was investigated. The error in the
shear stress is approximately a function of , for angles less than 20 degrees. Dye tests( cos )1− α
were used to show that the maximum angularity of the flow was 20 degrees. For = 20o the errorαin measured shear stress is approximately 6 percent. For additional information about the testing
procedures, the interested reader is refereed to Ippen et al. (1960, 1962a, 1962b).
The existence of a fully developed boundary layer at the upstream end of the bend is highly
desirable. A boundary layer which is still developing through the test section may yield variations
in shear stress which are in part an artifact of the evolving boundary layer. The boundary layer
thickness can be approximated by the Blasius expression if one assumes turbulent flow from the
channel entrance at x = 0,
(6)δ
υx Vx
=
0 3815
.
For a kinematic viscosity, , of 9.29 x 10-6 m2/s (10-5 ft2/s) and a distance of x = 6.096 m (20 ft)υ
from the inlet to the bend entrance, the boundary layer thickness, , is 116.8 mm (4.6 inches) forδ
an average channel velocity, V, of 0.427 m/s (1.4 ft/s). Therefore, the boundary layer was fully
developed for flow depths of 50.8, 76.2, and 101.6 mm (2, 3, and 4 inches), and nearly developed
for depths of 127 and 152.4 mm (5 and 6 inches).
The achievable accuracy of the energy gradient measurements was not sufficient to
determine the existence of uniform flow. However, Ippen et al. (1962b), suggest that the existence
of uniform flow is not critical to the investigation. Experimental data was used to demonstrate that
the distribution of relative shear stress, , is little affected by variations in the approach flow,τ τo o/
Froude number, and energy gradient.
Flume Data Measurements
Measurements of local velocity were taken for a range of flow rates at the ten stations shown
in Figure 1. Additional data was collected between stations 7 and 8 due to highly non uniform flow
patterns. Ippen et al. (1960), took readings approximately every two inches across each station for
the shear measurements except in the outer four inches of the wetted perimeter where no data was
collected.
Velocity measurements were made using a 7.938 mm (5/16 inch) Prandtl tube. The
instrument was always aligned parallel to the downstream flow direction. No attempt was made to
determine the cross stream velocity components. Sufficient velocity measurements were taken to
determine the gross characteristics of the velocity field (Ippen et al., 1962b).
The water surface elevation or depth was measured using a point gauge mounted on the
instrument carriage. Additionally, water depth was at times measured using static pressure
measurements. Sufficient water surface elevation measurements were collected to properly define
the super elevation of the flow (Ippen et al., 1962b).
Data Presentation
Ippen et al. (1960, 1962a, 1962b) present data related to water velocity, water surface
elevation and shear stress measurements. Water velocities are presented in the form of velocity
contour maps at each station, while water surface elevations are presenteed as cross section plots
showing the depth at each station. The objective of the research by Ippen et al. (1962b) was to
determine the shear stress distribution in bends. The shear stress at each station, and contour maps
showing the shear stress distribution are shown. Making plots for three variables at 10 stations for
6 runs yields 180 plots in addition to the 6 contour maps of shear distribution. The large number of
plots required that Ippen et al. (1960, 1962a, 1962b) limit the results which were published.
Combining the information published in all three references, water depth is presented at each cross
section for the 76.2 and 101.6 mm (3 and 4 inch) deep tests, shear plots are made for each cross
section along with shear contour maps for the 76.2, 101.6, 127, and 152.4 mm (3, 4, 5, and 6 inch)
deep tests. Velocity contour maps are presented for the 76.2 mm and 152.4 mm (3, and 6 inch) test.
Water velocity and depth are reported in dimensional form, while the shear stress was non-
dimensionalized by dividing the measured values with the average shear stress of Station 1.
Numerical Model
A numerical model of the 76.2 mm (3 inch) deep flume experiment was created to assess the
ability of a Computational Fluid Dynamic (CFD) program to simulate a complex three dimensional
flow field with a free surface. The commercially available software packages Gambit and Fluent
were used to generate the computational mesh and solve the flow field respectively. The flume
geometry in the numerical model was identical to that reported by Ippen et al (1962b). To include
the effects of super elevation in the bend a two fluid (two phase) model was used, the area above the
trapezoidal section was extended to include a pocket of air. A structured grid with 166,100
computational cells was applied to the flow field. Grid resolution was varied to throughout the flow
field to concentrate computational cells in areas of particular interest. A 5 cell �boundary layer grid�
normal to the flume walls was created to improve resolution of the velocity gradient. The cell
closest to the wall has a thickness of 1.27 mm (0.05 inches) and the total boundary layer grid
thickness is 11.48 mm (0.452 inches). The cell thickness in the boundary layer increases
exponentially with the distance from the wall such that the second cell is 1.3 times the thickness of
the first. Figure 2 shows a cross section of the channel used for the modeling. A total of 20 cells
are used in the vertical direction below the estimated free surface. In the lateral direction, 45 cells
were used below the free surface. In the downstream direction, 110 cells were used, with a typical
cell length of 76.2 mm (3 inches) between Stations 1 and 10.
Model Options
Numerous options are available in the Fluent software package to model the described flow
field. It is beyond the scope of this paper to discuss all of the options available in Fluent, however,
relevant aspects of the model are presented.
Direct solution of the Navier-Stokes equations (Direct Numeric Simulation) is not feasible
at this time for the flow field of interest. Therefore, a turbulence model for viscous energy
dissipation is required. The seven equation Reynolds Stress turbulence model was used for viscous
energy dissipation with standard wall functions. The Reynolds Stress Model (RSM) does not utilize
an isotropic eddy viscosity, rather it solves transport equations for the Reynolds Stresses in
conjunction with an equation for dissipation rate. The RSM does not always yield results which are
clearly superior to those of the simpler models unless flow features result in anisotropic Reynolds
Stresses. (Fluent, 1998). However, for this application, the stress induced secondary currents in the
model suggest the use of the RSM.
The standard wall functions used with the RSM in Fluent are based on work by Launder and
Spalding (Fluent, 1998). The law of the wall for mean velocity yields:
U Ey* *ln( )=1κ
(7)
where
UU C kP P
w
*/ /
/≡ µ
τ ρ
1 4 1 2
(8)
yC k yP P*
/ /
≡ρ
µµ1 4 1 2
(9)
and k = von Karman constant (0.42), E = empirical constant (9.81), Up = mean velocity of the fluid
at point P, kp = turbulent kinetic energy at point P, yP = distance from point P to wall, µ = dynamic
viscosity of the fluid and Cµ = constant (0.09). The logarithimic law for mean velocity is known to
be valid for y* > 30 ~ 60 and is used in Fluent for y* > 11.225. When the mesh is such that y* <
11.225 then the laminar stress-strain relationship U* = y* is used. Figure 3 shows the predicted
values of y* for the grid used in the investigation when the Reynolds Stress model is applied.
Fluent offers the user a choice of the numerical scheme used in the solution of the governing
equation. Fluent uses a �finite-volume� approach for discretization of the governing equations. A
segregated solver was selected whereby the non-linear governing equations are solved sequentially
in an iterative process. The governing equations are linearized implicitly. A second order upwind
solution was used for the momentum equations, the Reynolds Stress equations, the volume fraction,
turbulent kinetic energy, and turbulent dissipation rate. The PISO scheme was used for pressure-
velocity coupling due to the degree of skewness in the grid in the corners between the bottom and
sides of the flume. Finally, the PRESTO scheme was used for pressure interpolation as
recommended by Fluent for multi-phase flows. The multi-phase flow computations were handled
using an implicit Volume of Fluid approach. For a more detailed explanation of the solution
schemes, the reader is referred to the Fluent 5 Users guide (Fluent, 1998).
The iterative nature of the solution schemes employed requires that convergence criteria be
specified for terminating the iterations. Convergence was determine when the average shear stress
at Station 1 and 7 was no longer changing from one iteration to the next.
For the purpose of comparison, the model was also run with the �standard� two equation k-
epsilon turbulence model. The k-epsilon model also used standard wall functions as described
above and all of the same solution schemes were used. The k-epsilon model is popular for
engineering applications and receives widespread application. In this application, it is of particular
interest to use the k-epsilon model for comparison since it utilizes an isotropic eddy-viscosity and
would be expected to produce different results where secondary currents are present.
Boundary conditions
The downstream boundary condition was two pressure outlets, one for water and the other
for air. The water outlet extended from the flume bottom to the water surface elevation set by Ippen
et al. (1962a). A user defined function was used to describe a hydrostatic pressure distribution over
the outlet. The upper pressure outlet (air) was set at atmospheric pressure. Fluent does allow for
the use of an �outflow� boundary which determines the flow field properties at the boundary from
the interior flow field, however, and outflow boundary does not handle reverse flow correctly since
this requires information from outside the flow field. Though reverse flow does not occur in this
example when the final solution is found, during the iterative solution process, it was noted the
reverse flow would occur periodically.
The upstream boundary condition was comprised of two velocity inlets, one for water and
one for air. The water inlet extended from the bottom of the flume the water depth which Ippen et
al (1962b). observed at Station 1. The velocity of the water was set such that the volumetric flow
rate matched that of the physical experiment. The velocity of the air was set the same as that of the
water to avoid inducing any wind shear. No attempt was made to apply a velocity profile to the
inflowing water. Similar to the physical experiment, the velocity profile was allowed to develop
between the inlet and Station 1.
The walls of the flume were set hydraulically smooth, matching the roughness used by Ippen
et al. (1962b). The top of the flume (in contact with air only) was set as a symmetry boundary,
making the flow field symmetrical about the top of the flume. In this application, the use of a
symmetry boundary on the top of the model gives a frictionless boundary.
Results
Model results are presented in the same manner used by Ippen et al. (1962a, 1962b). Figures
4a and 4b show the measured water surface elevation at Stations 1 through 10 along with the
predicted water surface elevations using the Reynolds Stress model and the K-epsilon model. The
values for the measured data were obtained by digitizing the plots shown in Ippen et al. (1962a).
Error bounds of plus or minus 2.5 percent of the measured value are shown. The units used for the
plots are US customary rather than SI to remain consistent with the original published plots. In
general there is good agreement between observed values and the predicted water surface elevations
for both turbulence models. Several specific observations about the water surface elevations are
made:
1) Both numerical models over predict the depth at the upstream end and under predict the
depth at the downstream end, indicating a steeper water slope than the measured water
surface slope. Ippen et al. (1962b) reports that during the study, the slope of the flume
decreased from 0.000 64 to 0.000 55. The change in slope is a possible source for a portion
of the discrepancy. Over the reach length a difference of 0.5 mm (0.02 inches) would result.
However, it is thought that most of the difference is the result of difficulties and
uncertainties in establishing the downstream boundary condition in the physical model.
2) The Reynolds Stress model clearly shows more super elevation in the outside of the bend
at Stations 4-7, giving closer agreement to the observed values. At Stations 8-10 the K-
epsilon model shows a depression in the water surface elevation on the outside of the bend
and some super elevation on the inside of the bend. This is inconsistent with the observed
values. The Reynolds Stress model more accurately reproduces the water surface profile
observed, showing relatively flat water surface elevations at Stations 8-10.
3) The difference in the water surface elevations predicted by the two turbulence models is
consistent with the limitations of the turbulence models. The isotropic eddy viscosity used
in the k-epsilon model dampens the strength of the secondary currents and consequently the
magnitude of the super elevation.
The predicted and observed shear stresses for Stations 1-10 are shown in Figures 5a and 5b, while
Figure 6 shows a plan view of the predicted shear stress distribution in the entire flow field. For
comparison, the shear stress map produced by Ippen et al. (1962a) is reproduced in Figure 7.
Contour intervals in Figures 6 and 7 are the same. Ippen et al. (1962a) measured the shear stress
at approximately 2 inch intervals (11 locations) across the bottom of the flume and made no
measurements within the outside 4 inches. The shear stress curves which Ippen et al. (1962a) show
were digitized and are also shown in the Figures 5a and 5b. Based on the experimental descriptions,
the estimated location of the measured values are shown. All plots show relative shear on the
vertical axis. Relative shear is computed as the local shear stress divided by the average shear stress
of Station 1. The horizontal axis is again shown in US customary units rather than SI units to remain
consistent with the original published plots. Analysis of the model results and comparison with the
measured values yields the following observations:
1) The predicted average shear stress at Station 1 using the Reynolds Stress model was 0.331
Pascal (69.2 x 10-4 psf), while the average shear stress reported by Ippen et al. (1962b) was
0.335 Pascal (70 x 10-4 psf).
1) For Station 1, Ippen showed a plot (Ippen et al., 1962a) showing the actual
nondimensionalized shear stress measurements. Ippen indicated that he believes the shear
stress at Station 1 is symmetrical about the centerline of the flume. By plotting the actual
values observed by Ippen et al. (1962a) and the mirror image of the values, the uncertainty
in the measurements becomes apparent. Variations between mirrored points is typically on
the order of 10 percent. Error bars shown on all of the plotted data points are therefore plus
and minus10 percent.
2) All of the plots show a significant decrease in the predicted shear stress in the corner
between the flume bottom and the side wall when the Reynolds Stress model is used. The
decrease is noted where the boundary layer of the side wall and the bottom intersect, a
location where it would have been impossible to obtain any physical measurements due to
the diameter of Prantle tube used in the experiments. It is not clear why the k-epsilon model
frequently shows an increase in the shear stress in the corner and no explanation is offered,
beyond the observation that there is no physical basis for the increase and this may be a short
coming of the k-epsilon model.
3) At Stations 1 through 5, the Reynolds Stress and k-epsilon model produce very similar shear
stresses. The lack of secondary currents in the upstream portion of the bend clearly shows
the similar performance of the two models when the assumption of an isotropic eddy
viscosity is valid.
4) At Stations 6 through 10, the two turbulence models produce very similar results in the left
half of the channel. However, in the right half of the channel, the k-epsilon model clearly
under predicts the shear stress. Values are significantly lower than those predicted by the
Reynolds Stress model. The development of secondary currents in the downstream portion
of the bend produces a condition where the assumption of an isotropic eddy viscosity is no
longer valid. At these sections, the Reynolds Stress model produces results which more
closely match the observed values. Figure 8 shows the velocity components in the plane of
Station 7, showing the presence of secondary currents.
5) At Station 5, 6, and 7 the Reynolds Stress model very closely matches the observed shear
stress values. At Station 8 however, the Reynolds Stress model accurately predicts the shape
of the shear stress distribution but tends to under predict the magnitude.
6) At Station 10, at a position of approximately +8 inches, Ippen et al. (1962a) measured a
substantial and abrupt increase in the shear stress. The numeric models show a fairly
constant shear stress from a position of 0 inches to +9 inches. The measured data is
inconsistent with observations at Station 9, and may be anomalous measurements.
Conclusion
A comparison of measured and predicted water surface elevations and shear stresses using
Computational Fluid Dynamics and flume measurements was made. For the CFD calculations, two
turbulence models were used, the k-epsilon model and the Reynolds Stress model. Results from the
two models were compared to each other and the flume data. Both numerical models produced
results which, if given no other information, would appear to be very plausible. However, a
comparison of the k-epsilon model results with the RSM results and the measured data, illustrates
the known shortcomings of the k-epsilon model in flow fields where strong secondary currents
occur. It is imperative that the modeler be aware of the limitations of the model being used,
reviewing only the results of one model may not reveal a shortcoming. The use of CFD by an
inexperienced user may result in an incorrect interpretation of a flow phenomena.
The Reynolds Stress model typically produced shear stresses within 10 percent of the
measured values. The measured distribution of the shear stress in a bend was matched, showing that
CFD can be used to predict shear stress magnitude and distribution under smooth, rigid boundary
conditions. Given the very high cost of obtaining quality flume data, CFD offers a cost effective
alternative, even when the licensing fees of the software are considered. Furthermore, a CFD model
ultimately produces far more data than a flume study, and can provide an insight on the behavior of
the flow which would be cost prohibitive to collect. Over 166,000 three dimensional velocity
�measurements� are given by the CFD model in this application, and over 5000 shear values were
reported in the reach of interest, by comparison, Ippen et al. (1962b) collected fewer than 200 shear
stress measurements in the same reach
Finally, it must be noted that use of CFD without any validation is not wise. Validation can
vary and does not imply that a physical model is required of each numerical model, however some
basis for believing the numerical model results should exist. Without the existence of physical data,
at a minimum a grid resolution study should be conducted to insure that results are not an artifact
of the computational grid.
References
Fluent Inc. (1998). �Fluent 5 Users Guide�, Volumes 1-5. Fluent Inc., Centerra Resources Park, 10
Cavendish Court, Lebanon, NH 03766.
Ippen, A.T., Drinker, P.A., Jobin, W.R., Noutsopoulos, G.K.. (1960). �The Distribution of
Boundary Shear Stress in Curved Trapezoidal Channels.� Massachusetts Institute of
Technology Hydrodynamics Laboratory Technical Report No. 43.
Ippen, A.T., Drinker, P.A., Jobin, W.R., Shemdin, O.H. (1962a). �Stream Dynamics and Boundary
Shear Distributions For Curved Trapezoidal Channels.� Massachusetts Institute of
Technology Hydrodynamics Laboratory Technical Report No. 47.
Ippen, A.T., Drinker, P.A. (1962b). �Boundary Shear Stress Stresses in Curved Trapezoidal
Channels.� J. Hydr Div.., ASCE, HY5 143-179.
Preston, J.H. (1954). �The Determination of Turbulent Skin Friction by Means of Pitot Tubes.�
J. of Royal Aeronautics Soc. Vol. 54, 109-121.
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Position
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
Wat
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leva
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(in)
RSMIppen curveK-E+ - 2.5 Percent
Station 1
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2.6
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Station 2
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Station 3
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Station 4
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Station 5
Figure 4a. Observed and predicted water surface elevations Stations 1 through 5.
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Position
2.6
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Station 6
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Station 7
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Station 8
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Wat
er S
urfa
ce E
leva
tion
(in)
Station 9
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
Wat
er S
urfa
ce E
leva
tion
(in)
Station 10
Figure 4b. Observed and predicted water surface elevations Stations 6 through 10.
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
Position
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
RSMIppen PointsIppen curveK-EIppen Mirror
Station 1
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 2
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 3
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 4
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 5
Figure 5a. Observed and predicted shear stress Stations 1 through 5.
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
Position
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 6
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 7
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 8
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 9
-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22
21.81.61.41.2
10.80.60.40.2
0
Rel
ativ
e Sh
ear
Station 10
Figure 5b. Observed and predicted shear stress Stations 6 through 10.