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Representation of Weirs in the CMS

By Christopher Reed and Alejandro Sánchez

Last Date Modified: September 17, 2010

Introduction

Weirs are a common coastal engineering structure used to control flow and can affect sediment transport. They are typically used in weir jetties or in wetlands to control discharges, provide flood control, act as salinity barriers and optimally distribute freshwater to manage salinity regimes and sedimentation rates and deposition patterns. Since these structures are a significant component of hydrodynamic and sediment transport controls in the coastal zone, it is important that the CMS simulate their effects. The simulation of weirs is based on the standard weir equation for either sharp-crested or broad-crested weirs, with a foundation in Bernoulli’s equation. The implementation of weirs in the CMS is validated in two applications on the lower Mississippi River. The weirs are applied in the simulation of flow over the Bonne Carrie spillway north of New Orleans.

Formulation

Two approaches are developed to implement weir structures in the CMS. In the first approach the standard weir equation for a rectangular crosssection is introduced in the model as follows (HEC 2010):

Q=C_{df}C_{w}Lh^{1.5}

(1)

where Q is the flow rate over the weir crest, $C_{w}$ is the weir coefficient, $L_{w}$ is the weir crest length, $h$ is the upstream water depth above the crest, and $C_{df}$ is the submergence correction factor (also referred to as the drowned flow reduction factor). A definition schematic for both the free flow and submerged flow conditions is provided in Figure 1.

Figure 1. Schematic showing weir flow conditions.

To obtain a non-dimensional weir coefficient, Equation (1) can be re-written as (Reed and Sanchez 2011)

Q=C_{df}C'_{w}{\sqrt {g}}Lh^{1.5}

(2)

where $C'_{w}=C_{w}/{\sqrt {g}}$ . Ranges for $C_{w}^{'}$ were suggested in HEC (2010). For a sharp-crested weir, it is between 0.55 and 0.58, and a broad-crested weir between 0.46 and 0.55.

The submergence coefficient (C_df) is determined using different methods. For a sharp-crested weir, the Villemonte formula is used,

C_{df}={\Bigg [}1-{\biggl (}{\frac {h_{2}}{h_{1}}}{\biggr )}^{1.5}{\Bigg ]}^{0.385}

(3)

where $h_{1}$ and $h_{2}$ are the upstream and downstream water levels above the weir crest elevation, respectively. For a broad-crested weir, $C_{w}^{'}$  is calculated as

C_{df}=\left\{{\begin{aligned}&\quad \quad \quad 1\quad \quad \quad \quad \quad \quad \quad \quad h_{2}/h_{1}\leq 0.67\\&1-27.8(h_{2}/h_{1}-0.67)^{3}\quad h_{2}/h_{1}>0.67\end{aligned}}\right.

(4)

For a spillway-type weir, $C_{d}f$ is obtained by fitting a curve to data (Reed and Sanchez 2011),

C_{df}=1-\exp {{\biggl (}-8.5{\frac {h_{d}}{h_{1}}}{\biggr )}}

(5)

Super-critical flow conditions occur when the tail-water elevation is sufficiently low. Under these conditions, the submergence coefficient, $C_{df}$ is equal to 1.0.

The second approach treats the structure cells as other internal cells by adding the x- and y-components of the resistance force terms $M_{x}{\text{ and }}M_{y}$ induced by the weir structures in the depth-averaged momentum equations. The resistance forcing is represented by a quadratic drag law and the Manning’s (n) needs to be specified as the drag coefficient.

Figure 2. Curve fit to the USACE EM 1110-2-1603, plate 3-5, Section A-A data.

Numerical Implementation

A specially designed implicit scheme is developed to couple the flow computations of the upstream and downstream of weir structures (Wu 2012). Multiple weirs can be specified in the CMS and each weir is implemented on a line of cells (cell string), as shown in Figure 3. The model assumes the same fluxes across the upstream and downstream interfaces of each weir cell, and the flux is calculated with the following equation that is modified from Equation (2):

q=C_{df}C_{w}^{'}C_{wl}{\sqrt {g}}h^{1.5}ds

(6)

where ds is the length of the face upstream or downstream of a weir cell, and C_wl is a coefficient of distribution of flow discharge over the cells of each weir structure. Since $\Sigma q=Q$ , the following constraint should be applied to the distribution coefficient, C_wl:

\Sigma C_{wl}ds=L_{w}

(7)

where the summation is applied over all the cells of each weir structure. If a constant C_wl is assumed, $C_{wl}=L_{w}/\Sigma ds$ . One may also specify different values for C_wl on the cells according to their locations, bottom elevations, etc. For example, a smaller value can be given to the cells near the banks, and zero for those cells where the weir is blocked or inactive.

The implicit CMS-Flow model uses the SIMPLEC algorithm to solve the continuity and momentum equations (Wu et al. 2011). The flux (q) in Equation (6) is calculated using an implicit scheme by expanding q to a first-order Taylor series and deriving the flux and water level corrections. In the simulations of salinity or sediment transport with weirs, the salinity and suspended sediment will be transported over weir structures, but the bed load will be trapped.

The CMS-Flow explicit solver uses a staggered grid for the basis of the numerical solution. Weirs are implemented on the cell faces by specifying the two adjacent cells. The order of the cell specification is not critical for the weir implementation since the algorithm implemented in the CMS will determine the flow regime and the flow direction from the specified weir crest elevation and the water elevation in the adjacent cells. Figure 6 shows a typical CMS grid with a weir specified at a cell face.

Figure 3. CMS grid and designation of weirs.

In addition to the weir location,the user must also specify the crest elevation relative to the model datum, crest length L and the coefficient QUOTE . Multiple weirs can be specified at each face and the total flow across the face is the sum of the flow calculated for each individual weir. The momentum flux associated with the flow over the weir (and subsequently across the CMS cell face) is included in both the mass balance and the momentum balance calculations. Also, a weir can be specified as additive or replacing the cell face momentum flux that is calculated using the CMS solver. When the replacing weir type is specified, the flow from all weirs associated with the cell face replaces the calculated momentum flux. If an additive weir is specified, then the final momentum flux for the cell face is calculated as the weighted sum of the weir and momentum equations fluxes:

Where and are the momentum equations shallow water and weir equations, is the total flow across the face, and the fractions and for the momentum equation and weir flow contributions are defined as:

The application of the additive weir type would be applicable when a weir is set in a levee or other elevated structure. In cases where the levee is over topped the flow across the levee would be a combination of the weir flow and the flow over the levee surface.

The total flux across the cell face will be the sum of all weirs specified for that cell face. However, only the fluxes associated with the weirs will be included in the momentum balance.

When salinity or sediment transport is included in simulations with weirs, the salinity and suspended sediment will be transported. The bedload components in sediment transport algorithms will not be transported over weirs.

Additional Information

This wiki technical note was prepared and funded under the Coastal Inlets Research Program (CIRP) being conducted at the U.S. Army Engineer Research and Development Center, Costal and Hydraulics Laboratory. Questions about this technical note can be addressed to to Dr. Christopher W. Reed (Chris_Reed@URSCorp.com) of URS Corporation, the CIRP Program Manager, Dr. Julie D. Rosati (Julie.D.Rosati@usace.army.mil), the assistant Program Manager, Dr. Nicholas C. Kraus (Nicholas.C.Kraus@usace.army.mil).

References

Buttolph, A. M., C. W. Reed, N. C. Kraus, N. Ono, M. Larson, B. Camenen, H. Hanson, T. Wamsley, and A. . Zundel, A. K. 2006. Two-dimensional depth-averaged circulation model CMS-M2D: Version 3.0, Report 2, sediment transport and morphology change. Coastal and Hydraulics Laboratory Technical Report ERDC/CHL-TR-06-7. Vicksburg, MS: U.S. Army Engineer Research and Development Center.

Hydrological Engineering Center (HEC). 2010. HEC-RAS River Analysis System, Hydraulic Reference Manual, Version 4.1, Davis, California: HEC, US Army Corps of Engineers.

Militello, A., Reed, C.W., Zundel, A.K., and Kraus, N.C. 2004. Two-Dimensional Depth-Averaged Circulation Model M2D: Version 2.0, Report 1, Technical Documentation and User’s Guide, ERDC/CHL TR-04-2, U.S. Army Research and Development Center, Coastal and Hydraulics Laboratory, Vicksburg, MS.

Reed, C. W., and A. Sanchez. 2011. Representation of rubble mound structures in the Coastal Modeling System. CIRP Wiki, http://cirp.usace.army.mil/wiki/References#Structures.

URS. 2009. Hydraulic Modeling of Proposed Container Facility Along Mississippi River at Bonnet Carre’ Spillway. URS Corporation, Tallahassee, FL.

Villemonte, J.R (December 25,1947). "Submerged Weir Discharge Studies." Engineering News Record, p. 866-869.

Wu, W., A. Sanchez, and M. Zhang. 2011. An implicit 2D depth-averaged finite-volume model of flow and sediment

transport in coastal waters, ICCE 2010. Vicksburg, MS: US Army Engineer Research and Development Center.

Wu, W. 2012. Implementation of structures in the implicit CMS2D model. An Interim Report to Coastal Inlets Research Program. Vicksburg, MS: US Army Engineer Research and Development Center.

Documentation Portal

Retrieved from "https://cirpwiki.info/index.php?title=Weirs&oldid=11610"

Category:

CMS-Flow

@@ Line 1: / Line 1: @@
 __NOTOC__
-<big>
 =='''Representation of Weirs in the CMS''' ==
@@ Line 12: / Line 11: @@
 ==Formulation==
-The Hydraulic Engineering Center (HEC, 2010) approach for calculating flow over rectangular weirs is adopted here.  The form of the weir flow equation as presented in HEC (2010) is:
+Two approaches are developed to implement weir structures in the CMS. In the first approach the standard weir equation for a rectangular crosssection is introduced in the model as follows (HEC 2010):
-        {{Equation| <math> Q = C_{df} C_w L h^{1.5} </math>|2=1}}
-where <math>Q</math> is the flow rate over the weir crest, <math>C_w</math> is the weir coefficient, <math>L</math> is the weir crest length, <math>h</math> is the upstream water depth above the crest, and <math>C_{df}</math> is the submergence correction factor (also referred to as the drowned flow reduction factor). A definition schematic for both the free flow and submerged flow conditions is provided in Figure 1.
+{{Equation|<math>Q = C_{df} C_w L h^{1.5} </math>|1}}
+where Q is the flow rate over the weir crest, <math>C_w</math> is the weir coefficient, <math>L_w</math> is the weir crest length, <math>h</math> is the upstream water depth above the crest, and <math>C_{df}</math> is the submergence correction factor (also referred to as the drowned flow reduction factor). A definition schematic for both the free flow and submerged flow conditions is provided in Figure 1.
 [[File:Weirs_Figure_1.png|thumb|left|500px|Figure 1. Schematic showing weir flow conditions.]]
 <br style="clear:both" />
-An inspection of Equation 1 reveals that the weir coefficient is dimensional. In CMS, the the equation is re-written as:
-        {{Equation| <math> Q = C_{df} C'_w \sqrt{g} L h^{1.5} </math>|2=2}}
-where <math>g</math> is the acceleration due to gravity and <math> C'_w = C_w \ \sqrt{g} </math>.  In this form, the coefficients are independent of the measurement system (i.e. Metric or English). The value for the weir coefficient depends on the upstream and downstream geometry of the weir.  Ranges for the coefficient suggested in HEC (2010) are listed in Table 1.0.
+To obtain a non-dimensional weir coefficient, Equation (1) can be re-written as (Reed and
+Sanchez 2011)
+{{Equation|<math>Q = C_{df} C'_w \sqrt{g} L h^{1.5} </math>|2}}
+where <math> C'_w = C_w / \sqrt{g} </math>.  Ranges for <math>C^'_{w}</math> were suggested in HEC (2010). For a sharp-crested weir, it is between 0.55 and 0.58, and a broad-crested weir between 0.46 and 0.55.
+The submergence coefficient (C<sub>df</sub>) is determined using different methods. For a sharp-crested weir, the Villemonte formula is used,
-<br style="clear:both" />
+{{Equation|<math>C_{df} = \Bigg[1 - \biggl( \frac{h_2}{h_1} \biggr) ^{1.5} \Bigg]^{0.385} </math>|3}}
-'''Table 1.0.  Suggested Weir Coefficient Values.'''
-{|border="2" cellspacing="0" cellpadding="4" width="40%" align="left"
+where <math>h_1</math> and <math> h_2 </math> are the upstream and downstream water levels above the weir crest elevation, respectively. For a broad-crested weir, <math>C_w ^'</math> is calculated as
-|align = "center"|'''Configuration'''
-|align = "center"| '''  Range*'''
+{{Equation|<math>C_{df} = \left\{
-|-
+\begin{align}
-|Sharp-crested
+&\quad \quad \quad 1  \quad\quad\quad\quad\quad\quad\quad\quad h_2/h_1 \leq 0.67 \\
-|align = "center"|0.55 – 0.58
+&1 - 27.8(h_2/h_1 - 0.67)^3 \quad h_2/h_1 > 0.67
-|-
+\end{align}
-|Broad-crested
+\right.
-|align = "center"|0.46– 0.55
+</math>|4}}
-|}<br clear="all">
-<nowiki>*</nowiki> <math> C_w </math> values in HEC (2010) converted to <math> C'_w </math>
+For a spillway-type weir, <math>C_df</math> is obtained by fitting a curve to data (Reed and Sanchez 2011),
+{{Equation|<math> C_{df} = 1 -  \exp{ \biggl(-8.5  \frac{h_d}{h_1} \biggr) }</math>|5}}
-Super-critical flow conditions occur when the flow is independent of the tail-water elevation.  These conditions occur when the tail-water elevation is sufficiently low that the sub-critical coefficient <math> C_{df} </math> is equal to 1.0.  For submerged conditions a different approach is used for setting      .  For sharp-crested weirs, the coefficient is based on the equation by Villemonte (1947):
+Super-critical flow conditions occur when the tail-water elevation is sufficiently low. Under these conditions, the submergence coefficient, <math>C_{df}</math> is equal to 1.0.
-        {{Equation| <math> C_{df} = 1 -  \biggl( \frac{h_2}{h_1} \biggr) ^{1.5} </math>|2=3}}
-where <math>h_1</math> and <math> h_2 </math> are defined in Figure 2.  For broad-crested weirs, the value of  <math> C_{df} </math> is calculated by fitting a curve to data from USACE EM 1110-2-1603 (plate 3-5, Section A-A).  An exponential curve was fit to the Section A-A data.  The fit is shown in Figure 2 and the equation for <math> C_{df} </math> is:
+The second approach treats the structure cells as other internal cells by adding the x- and y-components of the resistance force terms <math>M_x \text { and } M_y</math> induced by the weir structures in the depth-averaged momentum equations. The resistance forcing is represented by a quadratic drag law and the Manning’s (n) needs to be specified as the drag coefficient.
-        {{Equation| <math> C_{df} = 1 -  \exp{ \biggl(-8.5  \frac{h_d}{h_1} \biggr) }</math>|2=4}}
@@ Line 50: / Line 54: @@
 ==Numerical Implementation==
+A specially designed implicit scheme is developed to couple the flow computations of the upstream and downstream of weir structures (Wu 2012).  Multiple weirs can be specified in the CMS and each weir is implemented on a line of cells (cell string), as shown in Figure 3. The model assumes the same fluxes across the upstream and downstream
+interfaces of each weir cell, and the flux is calculated with the following equation that is modified from Equation (2):
+{{Equation|<math>q = C_{df}C_w ^' C_{wl} \sqrt{g} h^{1.5}ds</math>|6}}
+where ds is the length of the face upstream or downstream of a weir cell, and C<sub>wl</sub> is a coefficient of distribution of flow discharge over the cells of each weir structure. Since <math>\Sigma q = Q</math>, the following constraint should be applied to the distribution coefficient, C<sub>wl</sub>:
+{{Equation|<math>\Sigma C_{wl} ds = L_w</math>|7}}
+where the summation is applied over all the cells of each weir structure. If a constant C<sub>wl</sub> is assumed, <math>C_{wl} = L_w /\Sigma ds</math>. One may also specify different values for C<sub>wl</sub> on the cells according to their locations, bottom elevations, etc. For example, a smaller value can be given to the cells near the banks, and zero for those cells where the weir is blocked or inactive.
+The implicit CMS-Flow model uses the SIMPLEC algorithm to solve the continuity and
+momentum equations (Wu et al. 2011). The flux (q) in Equation (6) is calculated using an implicit scheme by expanding q to a first-order Taylor series and deriving the flux and water level corrections. In the simulations of salinity or sediment transport with weirs, the salinity and suspended sediment will be transported over weir structures, but the bed load will be trapped.
 The CMS-Flow explicit solver uses a staggered grid for the basis of the numerical solution.  Weirs are implemented on the cell faces by specifying the two adjacent cells.  The order of the cell specification is not critical for the weir implementation since the algorithm implemented in the CMS will determine the flow regime and the flow direction from the specified weir crest elevation and the water elevation in the adjacent cells.  Figure 6 shows a typical CMS grid with a weir specified at a cell face.
@@ Line 55: / Line 77: @@
 <br style="clear:both" />
-In addition to the weir location,the user must also specify the crest elevation relative to the model datum, crest length L and the coefficient  QUOTE      .  Multiple weirs can be specified at each face and the total flow across the face is the sum of the flow calculated for each individual weir.  The momentum flux associated with the flow over the weir (and subsequently across the CMS cell face) is included in both the mass balance and the momentum balance calculations.  Also, a weir can be specified as ''additive'' or ''replacing'' the cell face momentum flux that is calculated using the CMS solver.  When the ''replacing'' weir type is specified, the flow from all weirs associated with the cell face replaces the calculated momentum flux.  If an additive weir is specified, then the final momentum flux for the cell face is calculated as the weighted sum of the weir and momentum equations fluxes:
+In addition to the weir location,the user must also specify the crest elevation relative to the model datum, crest length L and the coefficient  QUOTE  .  Multiple weirs can be specified at each face and the total flow across the face is the sum of the flow calculated for each individual weir.  The momentum flux associated with the flow over the weir (and subsequently across the CMS cell face) is included in both the mass balance and the momentum balance calculations.  Also, a weir can be specified as ''additive'' or ''replacing'' the cell face momentum flux that is calculated using the CMS solver.  When the ''replacing'' weir type is specified, the flow from all weirs associated with the cell face replaces the calculated momentum flux.  If an additive weir is specified, then the final momentum flux for the cell face is calculated as the weighted sum of the weir and momentum equations fluxes:
@@ Line 74: / Line 96: @@
 ==References==
 *Buttolph, A. M., C. W. Reed, N. C. Kraus, N. Ono, M. Larson, B. Camenen, H. Hanson, T. Wamsley, and A. . Zundel, A. K. 2006. Two-dimensional depth-averaged circulation model CMS-M2D: Version 3.0, Report 2, sediment transport and morphology change. Coastal and Hydraulics Laboratory Technical Report ERDC/CHL-TR-06-7. Vicksburg, MS: U.S. Army Engineer Research and Development Center.
+*Hydrological Engineering Center (HEC). 2010. HEC-RAS River Analysis System, Hydraulic Reference Manual, Version 4.1, Davis, California: HEC, US Army Corps of Engineers.
 *Militello, A., Reed, C.W., Zundel, A.K., and Kraus, N.C.  2004.  Two-Dimensional Depth-Averaged Circulation Model M2D: Version 2.0, Report 1, Technical Documentation and User<nowiki>’</nowiki>s Guide, ERDC/CHL TR-04-2, U.S. Army Research and Development Center, Coastal and Hydraulics Laboratory, Vicksburg, MS.
+*Reed, C. W., and A. Sanchez. 2011. Representation of rubble mound structures in the Coastal Modeling System.  CIRP Wiki, http://cirp.usace.army.mil/wiki/References#Structures.
 *URS. 2009. Hydraulic Modeling of Proposed Container Facility Along Mississippi River at Bonnet Carre<nowiki>’</nowiki> Spillway. URS Corporation, Tallahassee, FL.
 *Villemonte, J.R (December 25,1947). "Submerged Weir Discharge Studies." Engineering News Record, p. 866-869.
+*Wu, W., A. Sanchez, and M. Zhang. 2011. An implicit 2D depth-averaged finite-volume model of flow and sediment
+transport in coastal waters, ICCE 2010. Vicksburg, MS: US Army Engineer Research and Development Center.
+*Wu, W. 2012. Implementation of structures in the implicit CMS2D model. An Interim Report to Coastal Inlets Research Program. Vicksburg, MS: US Army Engineer Research and Development Center.
 -----