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.

Figure 1. Plan view of Ippen’s experimental set up (Ippen et, al., 1960).

Figure 2. End view of channel section showing computational grid.

Figure 3. Y* values using Reynolds Stress model.

-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22

Position

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

Wat

er S

urfa

ce E

leva

tion

(in)

RSMIppen curveK-E+ - 2.5 Percent

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

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 2

-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 3

-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 4

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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 5

Figure 4a. Observed and predicted water surface elevations 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

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 6

-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 7

-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 8

-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 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.

Figure 6. Shear stress map from Reynolds Stress Model.

Figure 7. Shear stress map reproduced from Ippen et al. (1962a)

Figure 8. Velocity vector plot at Station 7 showing secondary currents.