1317

Spillway Discharge Coefficients 1317

REVISITING SPILLWAY DISCHARGE COEFFICIENTS FOR SEVERAL WEIR SHAPES

William Kortney Brown, E.I.T.1

Gregory S. Paxson, P.E.2

Bruce Savage, PH.D., P.E.3

ABSTRACT

For practicing engineers, spillway discharge is often estimated with the weir equation, using discharge coefficients obtained from hydraulics textbooks or other publications. These discharge coefficients are typically considered to be accurate and appropriate since they have been widely published and used for many years. However, in some cases, the sources for these values are more than 100 years old and there is little documentation of the experiments that were used to develop the discharge coefficients. Discharge coefficients for five weir shapes presented in Handbook of Hydraulics (Brater et al., 1996) are reevaluated through physical (flume) and Computational Fluid Dynamics (CFD) modeling and compared with the results published data. The results of the physical and CFD modeling are presented and compared with the historical data. This research suggests that applying the discharge coefficients published in Brater et al. may underestimate discharge for some of the weir shapes studied. The authors recommend further studies of these and other weir shapes. This study also demonstrates the value of practicing engineers collaborating with local Universities. The relatively inexpensive study allowed the consultant to obtain specific, valuable data while providing research credentials to the University and providing degree credits to the author.

INTRODUCTION

Weirs are common structures in dam, stormwater, and stream engineering. Design and/or evaluation of these structures often requires research to estimate the discharge coefficient used in calculating flows for a given upstream head. Many resources exist for these variables, and Handbook of Hydraulics (Brater et al., 1996) is considered one of the more widely referenced sources used by practicing engineers. This handbook provides data for numerous weir cross sections, including triangular, trapezoidal, several broad crested shapes, and selected irregular shapes. Brater et al. contains data from sources mostly compiled in the early to mid 1900s. Much of the research was conducted without the use of model verifications and there is limited documentation of model setups and assumptions. The purpose of this research is to perform additional modeling of selected weirs, to verify each experiment, and to compare the results with published data. The

1 Senior Staff Professional, Schnabel Engineering, West Chester PA 19382, [email protected] 2 Principal, Schnabel Engineering, West Chester PA 19382, [email protected] 3 Assistant Professor, Department of Civil and Environmental Engineering, Idaho State University, Pocatello ID 83209, [email protected]

1318 Innovative Dam and Levee Design and Construction

intent is not to replace existing research regarding the weir shapes in question; but rather to supplement the debate, question and/ or validate data that is commonly referenced by the hydraulic engineering community. For this study, five trapezoidal weir shapes were selected from Brater et al. based on discharge coefficient to head relationships considered suspect by the authors. A study utilizing both physical and numerical modeling was performed to research these weir shapes. Numerical modeling was performed at Idaho State University using the Flow3D computational fluid dynamics (CFD) program. Physical modeling was performed at Villanova Universitys teaching laboratory. One of the most commonly applied equations for computing discharge of overflow weirs was developed from experiments by James B. Francis (Horton, 1907). This equation is often aptly referred to as the Weir Equation, and is presented as Equation 1.

2/30LHCQ d= (1)

Where Q = discharge (cfs), Cd= discharge coefficient (English), L = length of the weir perpendicular to flow (ft), g = gravity (32.2ft/s2), h = piezometric head upstream of the weir, and v = velocity, and Ho = total upstream head (ft), which is defined below.

gv

hH2

2

0 += (2)

PURPOSE AND SCOPE

A literature review was performed prior to weir selection and modeling to provide basis for the proposed research. The following texts were reviewed;

1. Francis; Hydraulic Experiments 2. Horton Weir Experiments, Coefficients, and Formulas 3. Brater et al. Handbook of Hydraulics 4. Rouse and Ince; History of Hydraulics (1957) 5. Chow Open-Channel Hydraulics 6. Fritz, Hager, and Fellow Hydraulics of Embankment Weirs 7. Savage, Johnson, and Geldmacher Comparison of Physical Versus

Numerical Modeling of Flow Over Spillways 8. Alzalimehr and Bagheri; Discharge Coefficient of Sharp-Crested Weirs

Using Potential Flow *Two references considered most relevant to this study are summarized herein.

The literature review contained within Weir Experiments, Coefficients, and Formulas provides background on early hydraulic experimentation and the accuracy of these early experiments. The author presents the results of early experiments chronologically and shows the cumulative result of all of these experiments. One such result is documented as the East Indian Engineers Formula for thin edged weirs (often seen as the weir equation)


Q=CLH3/2. Interestingly enough, this equation is also cited as the Francis Equation; where Francis performed numerous experiments to determine that n was 1.47, but he adopted 3/2 or 1.5 to simplify it. Additionally, the term H is noted as velocity head; a common area of discrepancy in the hydraulic engineering community surrounds the term H because some take it to simply mean static head or piezometric head, while in the weir equation it is meant as total head. Throughout Hortons discussion of the results of Bazin, it appears that one of Bazins goals was to determine if and when a single discharge coefficient would be applicable for all heads for a given weir shape. Bazin presented correction factors for the weir equation to deal with geometric adjustments on the model. Bazins work becomes very complicated in presentation because he used two forms of the weir equation, one including effects of velocity head, and the other, not including these effects in the discharge coefficient for a given weir. The results of some of his experiments are duplicated within the same document. Results of several experiments the Experiments of United States Board of Engineers on Deep Waterways Performed at Cornell University are presented. In discussing the setup of the Cornell experiments, Horton stated that the experiments were performed with the use of a Standard Weir as the flow measurement device and hook gages to measure flow depth. Geometry and construction of the Standard Weir is not discussed, though the rating curve for the Standard Weir was obtained from Bazins experiments. The fact that the Cornell experiments flow rates are based on the results of a separate but similar weir experiment may have led to compounded errors. If the Standard Weir experiment was not performed under the exact conditions as in the Cornell experiments, the use of this Standard Weir might have been inappropriate. A widely cited source for discharge coefficients in the hydraulic community is the collection contained in Brater et al. Handbook of Hydraulics. Referencing numerous private, academic and government based research projects, the collection of discharge coefficients in this text is considered by most to be plentiful. Some of the pertinent cross sectional (in the direction of flow) shapes published include trapezoidal, rounded edge, triangular, and sharp crested. While the authors did not perform the experiments resulting in the published values, their sources are prominent experimenters and engineers from the mid 19th century forward. There are, however, a number of irregularities in the discharge coefficient section that merit discussion yet without resolve, they continue to be one of the leading references of hydraulic engineers. One of the more controversial questions about these tables is that many of the values seemed to be relatively high. One might expect discharge coefficients in the mid to high 3s, or a little more than 4.0 for hydraulically efficient shapes. An example of this questionability is the simple triangular cross section (Table 5-8 in Brater et. al 1996). Performance is supposedly in the range expected of some ogee shaped weirs. Another anomaly is the fact that many of the discharge coefficients seem to increase without bound as the limits of the study are approached. Surely, one cannot expect this trend to continue, and one might wonder if the upper limits were simply extrapolated from a presumed functional relationship when the reality may be much different. This circumstance is seen in the


making of table 5-6 (Brater et al., 1996). Another anomaly seen in these tables is the occurrence of constant discharge coefficients for increasingly greater heads. For instance, in table 5-11 (Brater et. al, 1996), the weir with a 5H:1V upstream slope shows no change in discharge coefficient for a wide change in head. A typographical error is noted in Brater et al. under the discussion of table 5-9 (Brater et. al, 1996). Instead of stating that Bazin performed tests on weirs of height 2.46, the author states a height of 2.64. Brater et al. earlier discuss testing by Bazin on triangular weirs of height 2.46, and according to Horton, this trapezoidal series was performed with a weir height of 2.46. Figure 1 illustrates the Cd to Ho/P relationships presented in Brater et al. (1996) for the five weir shapes selected for evaluation.

Figure 1. Selected Discharge Coefficient Relationships for Selected Triangular and

Trapezoidal Weirs from Brater et al. (1996) It is important to note that the sizes shown in Brater et al. could not be accommodated in the lab space available; therefore, scaling was used for the physical modeling portion of this study. Additionally, due to flow constraints, not all heads included in Brater et al. could be modeled. In general, an Ho/P ratio below 0.7 was obtained during physical modeling. Froude scaling was considered to adjust modeling results; however the large scale of models reasonably negates scale effects. 1) Table 5-8 (Brater et al., 1996): Triangular XS with 1H:1V upstream & downstream slopes. Bazin conducted the experiments at Cornell University, with a height of 1.64. A weir of this height could not be accommodated in the available flume; therefore, a scale ratio of 1:3.28 was used for the physical modeling. Figure 2 shows the weir dimensions used for this study.

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

4.1

4.3

4.5

0 0.2 0.4 0.6 0.8 1

Dis

char

ge C

oeffi

cien

t (C

d)

Total Head over Weir Height (Ho/P)

Triangular 1:1 US and DSTrapezoidal 1:2 US and 1:1 DS With CrestTrapezoidal 2:1 US and Vert. DS With CrestTrapezoidal 1:5 US and Vert. DS With CrestTrapezoidal Vert. US and DS With 1:6 Inclined Crest


Figure 2. Triangular 1:1 Upstream and Downstream Slope Weir

2) Table 5-9 (Brater et al., 1996): Trapezoidal XS with 1H:2V upstream and 1H:1V downstream slope. Bazin conducted the experiments referenced in Brater et al., with a height of 2.46ft and crest width of 0.66ft. A weir of this height could not be accommodated in the available flume; therefore, a scale ratio of 1:5.28 was used for the physical modeling. Figure 3 shows the weir dimensions used for this study.

Figure 3. Trapezoidal 1H:2V Upstream Slope and 1:1 Downstream Slope Weir with

Crest 3) Table 5-11 (Brater et al., 1996): Trapezoidal XS with 2H:1V upstream slope and vertical downstream face, crest width=0.33ft (H= 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 (ft)). The United States Deep Waterways Bureau conducted the experiments referenced in Brater et al., with a height of 4.9ft and a crest of 0.33ft. A weir of this height could not be accommodated in the available flume; therefore, a scale ratio of 1:9.8 was used for the physical modeling. Figure 4 shows the weir dimensions used for this study.

Figure 4. Trapezoidal 2H:1V Upstream Slope and

Vertical Downstream Slope Weir With Crest

4) Table 5-11 (Brater et. al, 1996): 5H:1V upstream slope (H= 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 (ft)) The United States Deep Waterways Bureau conducted the experiments referenced in Brater et al., with a height of 4.9ft and a crest length of 0.66ft. A weir of this height could not be accommodated in the available flume; therefore a scale ratio of 1:9.8 was used for the physical modeling. Figure 5 shows the weir dimensions used for this study.

1

1

116"

FLOW

1

1

1

26"

FLOW 1.5"

12

FLOW 0.404"

6"


Figure 5. Trapezoidal 5H:1V Upstream and Vertical Downstream Slope Weir with Crest

5) Table 5-13 and Figure 5-22 (Brater et. al, 1996), (H= 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 (ft)) Bazin conducted the experiments referenced in Brater et al., with a height of 1.64. A weir of this height could not be accommodated in the available flume; therefore a scale ratio of 1:3.28 was used for the physical modeling. Figure 6 shows the weir dimensions used for this study.

Figure 6. Trapezoidal Vertical Upstream and

Downstream Slope Weir with 6H:1V Adverse Slope Crest

The final weir selected for the study is a sharp crested weir. The geometry for this model is described in Figure 7.

Figure 7. Sharp Crest Weir Section

PHYSICAL MODELING

The flume used for all of the physical modeling experiments is housed at Villanova Universitys Hydraulics Laboratory in John Barry Hall. The flume was leveled in the upstream/ downstream (flow) direction. A single 4 inch diameter ductile iron pipe (DIP) supplies water to the flume. A relatively new 4 inch Toshiba LF Combined Type electromagnetic flow meter and valve are used to record and adjust the flow entering the flume. A wooden head box was constructed by the author to contain and evenly distribute the flow from the supply pipe in the upstream most portion of the flume. At the outlet from the head box, a flow diffuser was used to quiet the outgoing flow. Each weir

15

FLOW0.81"

6"

16

FLOW

3.2"

6"

FLOW

0.220"

6"


was approximately 6 high, 28.88 inches wide and constructed of 0.220 thick plexiglass. The final height of each weir was measured after final fully leveled installation into the flume. A drawing of a typical test setup can be found in Figure 8.

Figure 8. Flume Profile

A water surface tip gauge was used to measure headwater upstream from the weir and the weir height. Discharge coefficient uncertainty resulting from tip gauge accuracy is taken into account in the results section. Head measurements were recorded after at least three minutes of a steady flow meter reading. The accuracy of a water surface tip gauge can be influenced somewhat by the user. Factors such as the meniscus on the tip, surface undulations or foam can influence the recorded value. To remain as consistent as possible, only the author took measurements. Additionally, a repeatability test was performed with a sharp crested weir to check for errors. During high flow situations, where there may be fast moving water, foam and slight surface undulations, an average water surface elevation was used.

NUMERICAL MODELING

To solve for the flow over the various weir shapes, the continuity equation and the momentum equations were solved numerically using a finite volume technique. The continuity and momentum equations were a modification of the commonly used Reynolds-average Navier-Stokes (RANS) equations with modified algorithms to track the free surface and model the weirs as a flow obstacle. To solve the RANS equations, a commercially available Computational Fluid Dynamics (CFD) code, Flow3D from Flow Science, Inc. was used. Because the weirs have a constant cross section in the lateral direction, they were numerically modeled as a sectional weir. Savage and Johnson (2001) showed that for dams with uniform cross sections, a 2-D analysis was sufficiently accurate to describe the 3-D flow and was computationally faster. Therefore, the flow field was discretized into a 2-D grid with a unit thickness in the lateral or y-direction. To find, define, and apply appropriate boundary conditions on a free surface, Flow-3D uses the Volume-of-Fluid (VOF) method (Hirt and Nichols, 1981). To define the weir structure within the grid, a grid porosity technique called the Fractional Area/Volume Obstacle Representation (FAVOR) algorithm was used. The FAVOR algorithm is outlined by Hirt and Sicilian (1985) and

4" DIP SUPPLY LINE

HEAD BOX

FLOW DIFFUSION DEVICE

FLOW

WEIR SPECIMEN

45 DEGREE BEND

6'-0"

3'-0" 3'-0"

WATER SURFACE TIP GAGE

3'-0"

FLOW METER AND 4" GATE VALVE UPSTREAM OF THIS POINT NOT SHOWN.


Hirt (1992). The FAVOR method is similar to the VOF method in that it also uses a first-order approximation to define the flow obstacle. This method eliminates the stair-stepping effect normally associated with rectangular grids and replaces all obstacle surfaces, curved or otherwise, with short, straight-lined segments. Although the VOF method updates temporarily to track the free surface, the FAVOR method is fixed and doesnt change with respect to time. To increase computational efficiency, a variable mesh was applied, using slightly longer grids in the x-direction away from the weir geometry. Variable grids allowed improved resolution in regions where f ow variables are rapidly changing and larger cells where resolution is not required. However, caution was exercised in using the variable grids because a rapid change in cells sizes can result in reduced numerical accuracy. This problem is outlined in Hirt and Nichols (1981). Multiple iterations were conducted to refine the mesh and initial conditions, and to ensure sufficient run time and acceptable runtime diagnostics to reach a steady state solution (tracked with flow convergence time history plots).

RESULTS This section includes comparisons between historic discharge coefficients from Brater et al. (1996), and raw data from the physical and numerical modeling portions of this research.

Figure 9. Triangular 1:1 Upstream and Downstream Slope Weir Data Comparison

The numerical modeling of the triangular weir shape developed discharge coefficients up to Ho/P of 3.04, whereas the Bazin and physical modeling results show data for Ho/P below 1.0.

3.50

3.75

4.00

4.25

4.50

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Cd

Ho/P

PHYSICAL M ODELING

BAZIN

NUMERICAL M ODELING


Figure 10. Trapezoidal 1H:2V Upstream and 1:1 Downstream Slope Weir with Crest

Data Comparison

In general, the results of testing for the trapezoidal 1H:2V upstream and 1:1 downstream slope weir with crest (Figure 10) show good agreement among the three sources. The difference between the physical modeling and the Bazin data show a consistent difference of about 7% to 8.5%, while the physical modeling and numerical modeling show differences of between 0.3% and 4%.


Data Comparison

The physical modeling for the study in Figure 11 showed that the discharge coefficient increases linearly from an Ho/P of 0.094 with a Cd of 3.71 to an Ho/P of 0.66 with a Cd of 4.33. The US Deep Waterways Board data, as presented in Brater et al. (1996), described a decreasing discharge coefficient for increasing Ho/P values. The numerical modeling

2.50

2.75

3.00

3.25

3.50

3.75

4.00

4.25

0.00 0.25 0.50 0.75

Cd

Ho/P

PHYSICAL MODELING

BAZIN

NUMERICAL MODELING

3.25

3.50

3.75

4.00

4.25

4.50

0.00 0.25 0.50 0.75 1.00 1.25

Cd

Ho/P

PHYSICAL M ODELING

US Deep Waterways Board

NUMERICAL MODELING


developed a slightly increasing discharge coefficient that approaches an asymptotic value of 3.87 for Ho/P values above 0.81.


Data Comparison

The physical modeling in figure 12 shows that the discharge coefficient increases linearly from an Ho/P of 0.092 with a Cd of 3.55 to an Ho/P of 0.70 with a Cd of 3.93. The US Deep Waterways Board data as presented in Brater et al. (1996) appears to show a decreasing discharge coefficient as the Ho/P value increases up to an Ho/P value of 0.31 where Cd remains constant at 3.39 for the remainder of the Ho/P values tested. The numerical modeling appears to show that the discharge coefficient increases from an Ho/P of 0.41 and Cd of 3.51, to an Ho/P of 1.02 and Cd of 3.61.

Figure 13. Trapezoidal Vertical Upstream and Downstream Slope Weir with 6H:1V

Adverse Slope Crest Data Comparison

3.25

3.50

3.75

4.00

4.25

0.00 0.25 0.50 0.75 1.00 1.25

Cd

Ho/P

PHYSICAL M ODELING

US Deep Waterways Board

NUMERICAL M ODELING

3.25

3.50

3.75

4.00

4.25

0.00 0.25 0.50 0.75

Cd

Ho/P

PHYSICAL MODELING

Cornell University

NUMERICAL MODELING


In Figure 13 the physical modeling data and Cornell University data show a difference of Cd of 0.5 for Ho/P from 0.02 to 0.37. At Ho/P of 0.37, the error from the Cornell study to the Physical modeling is +12%. The numerical modeling results show good correlation with the Cornell study for Ho/P from 0.04 to 0.13. Effects of Nappe Aeration For the trapezoidal 2H:1V upstream and vertical downstream slope weir with crest shape, heads between Ho/P ratios of between 0.38 and 0.5 were duplicated with aerated and non-aerated nappes. In general, the effect of a non-aerated nappe is an increased discharge coefficient. The percent increase in the discharge coefficient for this weir shape having a non-aerated nappe (for the three overlapping data points) was at most 2.3% for Ho/P ratios between 0.38 and 0.49. Physical Model Validation A sharp crested weir was tested in the same manner as the research models to validate the test setup of the physical modeling. The results are compared to the Rehbock and Swiss Society formulas, equations 3 and 4 respectively, for sharp crest weir discharge coefficients from 1912.

PH

HC 428.0

56.060

1235.3 +

+= (3)

+

+

+=2

2

5.0149.08.92

1288.3

dH

HC (4)

The sharp crest weir data shows good correlation between the physical modeling data, the Rehbock equation, and the Swiss Society equation for Ho/P below 0.3. Above Ho/P of 0.3, the Rehbock equation and the Swiss Society Equations deviate from one another. The Swiss Society equation is 12.9% higher than the Rehbock equation at Ho/P of 0.68. The physical modeling better follows the Swiss Society equation, with up to an 11% difference at Ho/P of 0.368. It is important to note that the physical modeling showed a change from an aerated nappe to a non-aerated nappe at Ho/P of approximately 0.368, and that the first few data points in the physical modeling study had clinging nappes. Uncertainty Analysis Certain variables within the physical modeling portion of this study had measurable accuracy. An example of this is the accuracy of the point tip gage used to measure weir height, water surface height, and floor height. To estimate the significance of these inaccuracies, an uncertainty analysis was performed. The uncertainty analysis is based on methods described in First-Order Uncertainty Analysis of an NPS Loading Model (Chadderton). An uncertainty curve was developed for all of the weir shapes, and generally showed that as the head increases, the uncertainty decreases seemingly reaching an asymptote of uncertainty towards the upper limits of the Ho/P axis. One of the reasons


for the asymptotic relationship is that the constant uncertainty of the piezometric head measurement becomes proportionally smaller to the head measurement as head increases. For Ho/P greater than approximately 0.15, the total uncertainty is below 1%, and for Ho/P greater than approximately 0.35, the total uncertainty is below 0.5%. Below Ho/P of approximately 0.15, the total uncertainty is as much as 4%. Repeatability Study To check the repeatability of the physical modeling portion of the study, two separate tests of a sharp crest weir were performed and recorded. The results of the test indicated good general repetition aside from some outlying data points particularly for lower heads. The test showed that the scatter of the points within each test is greater than the scatter between the two runs; therefore, the tests were repeatable.

CONCLUSIONS AND RECOMMENDATIONS

Historic Data and Conclusions Results of the physical and numerical modeling indicate that the historic discharge coefficient data published in Brater et al. may generally show underestimates of the discharge coefficients for many of the weir shapes and Ho/P ratios selected for this research, contrary to the authors initial theory. Table 1 shows best fit equations for the Brater et al. discharge coefficient data. Polynomial functions were generally selected because the R2 values were generally better than other available equations, and about the same as higher order polynomials

Table 1. Best Fit Equations for Brater et al. Discharge Coefficient Data and Corresponding R2 Values

Where x = Ho/P and y = Cd. Table 1 is based solely on Brater et al. data. Figure 14 shows the percent difference between the physical modeling and the Brater et al. discharge coefficient data best fit equations for each weir studied, and Figure 15 shows similar comparisons based on the numerical modeling from this study.

Numerical Data Best Fit Equation R2

y = 12.397x4 - 29.415x3 + 23.724x2 - 8.0766x + 5.1206 0.9927y = -0.2852x3 + 0.4958x2 - 0.498x + 3.9331 0.9989y = -3.4581x3 + 1.7732x2 + 2.369x + 2.4965 0.9992y = 3.39 1y = 54.652x3 - 34.177x2 + 4.9356x + 3.3621 0.9773

Weir ShapeTriangular 1:1 US. And DS.Trapezoidal 2:1 US. Vert. DS. With CrestTrapezoidal 1:2 US. 1:1 DS. With CrestTrapezoidal 1:5 US. Vert. DS. With CrestTrapezoidal 1:6 Crest with Vert. US. And DS.


Figure 14. Deviation of Physical Model Data from Brater et al. Discharge Coefficient

Data Best Fit Equations

Figure 15. Deviation of Numerical Modeling Data from Brater et al. Discharge

Coefficient Data Best Fit Equations

The vertical axis shows the Percent Difference between the Brater et al. discharge coefficient data best fit equation and the numerical modeling data, or the physical modeling data. Results show that the physical modeling was generally between 0% and 15% higher than the Brater et al. best fit equations, while the numerical modeling was generally between 5% of the Brater et al. best fit equations. Some shapes showed much closer relationships with the Brater et al. best fit equations than others. The physical and numerical modeling are not compared directly to the Brater et al. data because the while the Ho/P values for the physical and numerical modeling overlap the Brater et al. data, they were not sampled at the identical Ho/P values.

-25.0%

-15.0%

-5.0%

5.0%

15.0%

0.00 0.25 0.50 0.75Ho/P

Triangular 1:1 US. And DS. Trapezoidal 2:1 US. Vert. DS. With Crest

Trapezoidal 1:2 US. 1:1 DS. With Crest Trapezoidal 1:5 US. Vert. DS. With Crest

Trapezoidal 1:6 Crest with Vert. US. And DS.

-25.0%

-15.0%

-5.0%

5.0%

15.0%

0.00 0.25 0.50 0.75Ho/P

Triangular 1:1 US. And DS. Trapezoidal 2:1 US. Vert. DS. With Crest

Trapezoidal 1:2 US. 1:1 DS. With Crest Trapezoidal 1:5 US. Vert. DS. With Crest

Trapezoidal 1:6 Crest with Vert. US. And DS.


Physical and Numerical Modeling Conclusions Results indicate that the physical modeling was repeatable, but sensitive to measurement accuracy. This may be attributed to the limited experience of the author and the delicate nature of physical modeling. Previous research has shown good agreement between physical and numerical modeling measurements, and has shown that sensitivity is less of an issue for larger flumes. In this research, the agreement is less consistent. Table 2 shows the best fit equations selected for the numerical modeling results. Third order polynomial functions were selected because their R2 values were generally better than other available equations, and about as accurate as higher order polynomials.

Table 2. Best Fit Equations for the Numerical Data and Corresponding R2 Values

In Table 2, x = Ho/P and y = Percent Difference between the physical modeling data and the numerical modeling data best fit equation. Figure 16 shows the percent difference between the physical modeling data and the numerical modeling best fit equations for each weir studied. The physical modeling is not compared directly to the Numerical data because the Ho/P values for the physical modeling were not identical to the Ho/P values for the Numerical data.

Figure 16. Deviation of Physical Modeling Data from Numerical Modeling Best Fit

Equations With a few exceptions, the results of the physical modeling are from 100 to 114% of the numerical results. Thus, the physical modeling overestimates Cd (is biased high) if the numerical data is correct.

Numerical Data Best Fit Equation R2

y = 4.6502x3 - 7.9876x2 + 3.0438x + 3.7767 0.3838y = 0.511x3 - 1.3058x2 + 1.1246x + 3.5418 0.9991y = 0.0612x3 - 0.9437x2 + 2.2171x + 2.8478 0.9971y = 0.3632x3 - 0.9957x2 + 0.9954x + 3.2455 0.9998y = -0.3493x3 - 2.4247x2 + 0.4913x + 3.574 0.9411

Triangular 1:1 US. And DS.Trapezoidal 2:1 US. Vert. DS. With CrestTrapezoidal 1:2 US. 1:1 DS. With CrestTrapezoidal 1:5 US. Vert. DS. With CrestTrapezoidal 1:6 Crest with Vert. US. And DS.

Weir Shape

-10.0%

0.0%

10.0%

20.0%

0.00 0.25 0.50 0.75Ho/P

Triangular 1:1 US. And DS. Trapezoidal 2:1 US. Vert. DS. With CrestTrapezoidal 1:2 US. 1:1 DS. With Crest Trapezoidal 1:5 US. Vert. DS. With CrestTrapezoidal 1:6 Crest with Vert. US. And DS.


Nappe Aeration may influence the discharge coefficient by about 2% based on the limited data of this research. In general, an aerated nappe produced a lower discharge coefficient than a non-aerated nappe. Additionally, the first order uncertainty analysis of the physical modeling shows that for Ho/P greater than about 0.2, the total uncertainty explained by Cd uncertainty for the Trapezoidal weir with 1:2 upstream slope and 1:1 downstream slope with a crest, was less than 1%. With this conclusion, the numerical modeling shows a negative bias of between 0 and 14%, assuming that (except for 1% uncertainty) the physical modeling results are correct. Recommendations Because this research shows results different from the historic data, a thorough search for additional similar published weir data should be made. State of the art research facilities should be used to test the same weir shapes with geometry identical to the Brater et al. data. Similarly, the CFD modeling should duplicate the test conditions cited by Brater et al.. For the industry professional, caution is suggested when using historic Brater et al. discharge coefficient data. This research indicates that the historic information published in Brater et al. generally underestimates the discharge coefficient for many of the weir shapes and Ho/P ratios selected. Thus, the historic Brater et al. discharge coefficient data would generally underestimate the discharge capacity of the studied overflow weirs and Ho/P ratios selected. Alternatives for the historic Brater et al. discharge coefficient data can be found above, or adjustment factors may be deduced from Figure 14 and 15. This research also shows some very different Ho/P to Cd relationships for the studied weirs when compared with the Brater et al. data, which would significantly affect discharge rating curves. Similarly, this research shows that for some overflow weirs a discharge rating curve is necessary for accurate discharge calculation estimates, as opposed to using a constant discharge coefficient for all Ho/P values. In this research, the use of an uncertainty analysis highlights the critical variables. The use of uncertainty analysis techniques is recommended for all modeling and measurement studies. An additional suggestion for professionals comes from the manner in which this research was conducted. Combining a companys in-house talent with good relationships with local Universities can prove to be very fruitful. This research was conducted at Villanova University using a full-time working professional from Schnabel Engineering. The result is relatively inexpensive and targeted research for Schnabel, additional research credentials for Villanova University, and degree credits and targeted education for the author.


REFERENCES Afzalimehr, Hossein and Bagheri, Sara (2009) Discharge coefficient of sharp-crested weirs using potential flow, Journal of Hydraulic Research Vol. 47, No. 6 , pp. 820-823 Boitet et al. (2003) Hydrometry IHE Delft Lecture Note Series, Swets & Zeitlinger B.V., Lisse, The Netherlands Brater et al. (1982) Handbook of Hydraulics Sixth Edition, McGraw Hill, New York Crowe et al. (2000) Engineering Fluid Mechanics 7th Edition, John Wiley and Sons, Canada, pp. 616-619 Chadderton, First-Order Uncertainty Analysis of an NPS Loading Model Department of Civil and Environmental Engineering, Villanova University, Villanova, Pennsylvania Chow (1959) Open-Channel Hydraulics McGraw Hill Book Company INC., New York Ettema et al. (2000) Hydraulic Modeling ASCE Manuals and Reports on Engineering Practice No. 97, ASCE, Reston, Va Fenton, John and Zerihun, Yebegaeshet (2007) A Boussinesq-type model for flow over trapezoidal profile weirs Journal of Hydraulic Research Vol. 45, No.4, pp. 519-528 Francis (1868) Hydraulic Experiments Second Edition, D. Van Nostrand, London French, R. H. (1985) Open-Channel Hydraulics McGraw-Hill, New York. Fritz et al. (1998) Hydraulics of Embankment Weirs Journal of Hydraulic Engineering, 124(9), 963-971. Hirt, C.W. and Nicholes, B.D. (1981). "Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries." Journal of Computational Physics, Vol. 39, 201-225. Hirt, C.W. and Sicilian, J.M., (1985). "A porosity technique for the definition of obstaclesin rectangular cell meshes." Proceedings of the Fourth International Conference Ship Hydro., National Academy of Science, Washington, DC. Horton (1907) Weir Experiments, Coefficients, and Formulas Water Supply and Irrigation paper 200, Department of the Interior United States Geological Survey, Government Printing Office, Washington Houston et al. (1983) Hydraulic Model Study of Hyrum Dam Auxiliary Labyrinth Spillway GR-82-13, U.S. Department of the Interior, Bureau of Reclamation, Denver, Co.


(31 August 1992) Hydraulic Design of Spillways EM 1110-2-1603, U.S. Army Corps of Engineers, Washington, DC Kindsvater, C.E. (1964). Discharge Characteristics of Embankment Shaped Weirs U.S. Geological Survey Water Supply Kilpatrick and Schneider Use of Flumes in Measuring Discharge Chapter A14, Techniques of Water- Resources Investigations of the United States Geological Survey, USGS, Washington, DC Paper 1617-A, U.S., Government Printing Office, Washington, DC. Kobus (1984) Symposium on Scale Effects in Modelling Hydraulic Structures International Association for Hydraulic Research (IAHR), Esslinger am Neckar, Germany, September 3-6, 1984 Montes, S. (1998) Hydraulics of Open Channel Flow ASCE Press, Reston, Va Paxson et al. (2006) Labyrinth Spillways: Comparison of Two Popular U.S.A Design Methods and Consideration of Non-Standard Approach Conditions and Geometries international Junior Researcher and Engineer Workshop on Hydraulic Structures, The University of Queensland, Brisbane, Australia Rouse and Ince (1957) History of Hydraulics Iowa Institute of Hydraulic Research, State University of Iowa Savage, B.M. and Johnson, M.C. (2001). Flow over Ogee Spillway: Physical and Numerical Model Case Study. Journal of Hydraulic Engineering. Vol. 127, No. 8, Aug. pp. 640-649. Savage et al. (2001) Comparison of Physical versus Numerical Modeling of Flow over Spillways Utah State University Sharp (1981) Hydraulic Modeling Butterworth & Co Ltd, Boston Silyn-Roberts, Heather, 2005, Professional Communications, A Handbook for Civil Engineers, ASCE Press, American Society of Civil Engineers, Reston, VA, ISBN 0-7544-0732-0. Yakhot, V. and Orszag, S.A., (1986). "Renormalization group analysis of turbulence. I. Basic theory." Journal of Scientific Computing, 1(1), p 1-51.

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