6.2 Low Flow Through Bridges
Low flow is the most common analysis case for a bridge. Low-flow situations exist whenever the discharge passes through the bridge opening and the water surface or energy grade line elevations do not reach the elevation of the bridge low chord. A variety of potential solution methods is available in HEC-RAS for low flow, with low flow classified as Class A, B, or C, based on momentum computations at the bridge. Figure 6.3 illustrates the water surface profiles for each of the three classifications.
HEC-RAS must first determine the flow regime to properly classify the flow. It accomplishes this evaluation by computing the momentum at the bridge cross sections (2, BD, BU, and 3). First, HEC-RAS determines the momentum at critical depth at sections BD and BU for the given discharge. The section with the higher momentum is designated as the controlling section. If the two sections have equal momenta, section BU is selected as controlling.
For subcritical flow analysis, the momentum at section 2 is then computed and compared to the controlling section in the bridge. If the momentum at section 2 exceeds the critical momentum at the controlling section, the flow is assumed to be Class A throughout the bridge reach. If the momentum at section 2 is less than the critical momentum at the controlling section, Class B flow is assumed, with critical depth occurring within the bridge opening.
For supercritical flow, the momentum at section 3 is computed and compared to the critical momentum at the control section in the bridge opening. If the momentum at section 3 exceeds the critical value, the flow through the bridge is designated as Class C.
Equations for Low Flow
Modified from FHWA (HDS-1)
Figure 6.3 Low flow classification at bridges.
Energy Method.
The energy method, described in detail in Chapter 2, is applied in a similar manner for bridges. When an energy solution is obtained, losses through the bridge opening sections (2, BD, BU, and 3) are computed as if each were an unobstructed cross section. The friction losses between sections and losses due to expansion or contraction are computed and summed. If piers are present, the additional lengths on both sides of each pier are included in the wetted perimeter calculation and the area of the piers is removed from the cross-sectional flow area of sections BD and BU.
For bridge analysis with the energy method, expansion and contraction losses dominate through the bridge opening (between sections 2 and 3), as they are much larger than the friction losses through this same area. This is a reversal of the situation for normal valley cross sections. This is due to the short reach length between the upstream and downstream bridge face (minimizing the friction loss) and the (usually) large velocity heads experienced within the bridge opening, which often result in significant expansion or contraction losses. Water surface elevations and energy losses through a bridge are computed with Equations 2.43 through 2.46 (see (see page 68)), applying the standard-step method as if the bridge sections represented normal valley cross sections.
The energy method is also used to compute losses between sections 1 and 2 and between sections 3 and 4 for all of the other methods of bridge computations. Bridge computations differ only in the way the water surface elevations are computed between sections 2 and 3.
Momentum Method.
With the momentum method, momentum balances are computed through the four cross sections (2, BD, BU, and 3, respectively) that define the bridge opening. The equations used by the program for a momentum solution are presented in the following paragraphs.
From section 2, just outside the downstream face, to section BD, just inside the downstream face of the bridge, the momentum equation is written as
(6.1)
The forces Ff and Wx act in opposite directions and, when the distance between sections 2 and BD is limited, these forces are quite small compared to the other terms in the equation. These two terms are often neglected in hand computations, without significant error.
In HEC-RAS, Ff and Wx can be toggled on or off, together or independently. The default in HEC-RAS is for Ff to be included and Wx not. The Wx term requires an estimate of the channel slope, s0, between adjacent sections. Around bridges, s0 can be difficult to accurately determine and the slope may even be adverse (negative). In addition, the section just inside the bridge may have the same elevation as the section just outside the bridge, resulting in the value s0 = 0. Large errors in momentum can result from a poor estimate of the slope term.
The momentum equation, from section BD to section BU, is written as
(6.2)
From section BU to section 3 the momentum equation is
(6.3)
Drag forces are caused by the flow splitting around the piers, flowing along the piers, and then creating a downstream pier wake. The drag coefficient represents the effect of pier shape or streamlining. Common drag coefficients for piers are listed in Table 6.1.
The energy and momentum equations can both be used for Class A low flow at bridges with or without piers. If the water surface, or the energy grade line (if selected), exceeds the highest value of the bridge low chord elevation, the momentum solution is no longer valid. HEC-RAS reverts to a pressure flow, pressure/weir flow, or energy/weir flow solution. These combinations are discussed in Section 6.3.
Yarnell Equation.
The Yarnell equation (Yarnell, 1934) is another valid approach to examine Class A low flow through bridges with piers. The Yarnell equation is an empirical solution, developed in the 1920s from more than 2600 laboratory model tests. It evaluates the effect of bridge piers on the water surface elevation upstream of the bridge. The equation is most applicable for bridges that have many piers, with the piers causing the majority of the energy losses through the bridge. The Yarnell equation is concerned only with the pier shape, the pier obstructed area, and the velocity of the water. However, this method does not include any effects of the shape of the bridge opening, the shape of the abutments, or the width of the bridge. Yarnell's experiments were conducted for rectangular and trapezoidal channel shapes, so these shapes are most appropriate for application of the Yarnell method. Figure 6.4 shows a bridge that can be appropriately modeled with the Yarnell method. This bridge's width is about 300 ft (90 m) and there are 15 to 20 trestle bents supporting the roadway.
Figure 6.4 Flood flow passing through a railroad trestle bridge, St. Charles County, Missouri.
A railroad trestle is often best modeled with the Yarnell equation, as follows:
(6.4)
Only pier losses (no friction losses) are considered in Equation 6.4. H3-2 is simply added to the downstream water surface elevation of cross section 2 to obtain the water surface elevation at cross section 3, immediately upstream of the bridge. To use the Yarnell method, a K value must be assigned. This coefficient is based on the pier shape, as is the drag coefficient for the momentum method. Table 6.2 lists commonly used values for the Yarnell K.
Energy, momentum, or WSPRO may be more appropriate solutions for bridges for which significant losses are expected from bridge abutments or from the shape of the bridge opening. WSPRO is described in Section 6.12.
Class A Low Flow
Class A flow is the most common and corresponds to the Type I classification used by the FHWA. The flow regime is subcritical throughout the expansion and contraction reach, with an unsubmerged bridge opening. Figure 6.5 shows the I-255 bridge across the Mississippi River near St. Louis, Missouri, during the 1993 flood. Class A low flow occurred at this structure for all flows, including the peak discharge of 1,070,000 ft3/s (30,300 m3/s). For this flow condition, an adequate analysis of the bridge effect can be obtained by performing an energy or momentum analysis, by applying the Yarnell equation, or by using the methods developed by the FHWA for the WSPRO program.
Figure 6.5 I-255 crossing, Mississippi River near St. Louis, Missouri, June 1993.
Class B Low Flow
When the bridge opening width is small compared to the upstream width of flow at section 4, the opening may serve as a throttle. This situation can cause such a severe constriction of flow that critical depth occurs within or just downstream of the bridge opening, creating Class B flow (designated Type IIA or IIB in FHWA analyses). Under this type of flow pattern, the bridge acts as the control for the flow regime, and supercritical flow may be experienced (usually for a very short distance downstream of the bridge). Figure 6.3 shows a profile for Class B flow through a bridge. Class B flow may also occur in supercritical flow channels, potentially resulting in a hydraulic jump at the bridge. When critical depth occurs in the bridge, the Yarnell and the WSPRO methods are not appropriate, as both are based on subcritical flow. Rather, the calculations use either the energy or momentum equations. The momentum equation tends to be more appropriate, however, due to its ability to handle rapidly varied flow through the bridge. If the momentum equation doesn't converge to a solution, then energy methods are used.
For Class B low flow, the modeler should consider performing a mixed flow analysis, with both subcritical and supercritical computations to solve for the proper flow regime through the bridge. Chapter 8 discusses mixed-flow analysis procedures.
Class C Low Flow
At bridges crossing supercritical flow channels, Class C low flow (designated Type III flow by the FHWA) is the norm. Figure 6.3 displays a profile for Class C low flow. Bridges crossing steep, concrete-lined flood reduction channels or steep mountain streams can be designed for Class C flow. Both the energy and momentum methods are applicable in these cases, with the momentum equation preferred because of the high velocities and rapidly varied nature of supercritical profiles through bridges. Figure 6.6 shows a prismatic supercritical flow channel spanned by bridges in the Los Angeles, California area. For this constant cross-sectional reach, the cross-section geometry remains unchanged, so expansion or contraction reaches adjacent to the bridge are not needed. Only the piers affect the water surface elevations. The lack of an expansion or contraction reach is fairly typical in the analysis of most man-made channels. Note the long "splitter" extension on the bridge pier; this is added to smoothly split the flow and minimize pier losses through the bridge. Chapter 10 discusses this type of pier in more detail. Bridge effects in a Class C flow regime may require testing with a physical model to verify computational accuracy.
USACE
Figure 6.6 Pier extension in a supercritical flow channel, Los Angeles, California.
