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Publication Number: FHWAHRT07026 Date: February 2007 
Bottomless Culvert Scour Study: Phase II Laboratory ReportChapter 3: Theoretical BackgroundExperiments show that scour is generally deepest near the corners at the upstream entrance to the culvert. This observation is commonly attributed to the contraction (concentration) of flow near the upstream entrance of the culvert. Figure 4 illustrates the pattern of primary flow near this location, where water that is blocked by the embankments (in the approach to the culvert) is forced through the culvert opening. The vortices and strong turbulence just downstream of the culvert inlet, generated by the contraction of flow and typically called secondary flow, occur in the socalled separation zone. This flow pattern is very similar to the abutment scour phenomenon that researchers have observed for bridge scour.
Several researchers, including Chang, GKY and Associates, Inc., and Sturm, have suggested that bridge abutment scour can be analyzed as a form of flow distribution scour by incorporating an empirical adjustment factor to account for vorticity and turbulence.(2, 3, 4) The adjustment factor to account for vorticity and turbulence can be derived from laboratory results. These notions were used to formulate the theoretical background for analyzing the culvert scour data. Variables used in the data analysis are illustrated in the following definition sketches for unsubmerged (figures 5 and 6) and submerged (pressure) (figures 7 and 8) flow conditions. The notations in these figures are defined after the last figure.
CLEAR WATER SCOUREquation 1 is an expression for the unit discharge for an assumed flow distribution that remains constant as the scour hole develops. If no sediment is being transported into the scour hole, as was the case with all of our experiments, then no sediment can be transported out of the scour hole at equilibrium. In this case, the local velocity must be reduced to the critical incipient motion velocity, V_{c}, for the sediment size at the equilibrium flow depth, y_{2}. This equation forms the basis for the analysis:
where:
Note that the term on the left side of the equation is the assumed representative unit discharge across the scour hole at the beginning of scour, or q_{R}. Equation 1 can be rearranged to yield an equilibrium flow depth, y_{2}, once the representative velocity, V_{R}, and the critical incipient motion velocity, V_{C}, have been determined. This equilibrium depth reflects the scour that is attributed to the incoming flow distribution. The next two subsections will illustrate several ways to calculate the representative velocity and critical velocity. The third and fourth subsections will then discuss two different adjustments to the equilibrium clear water scour depth. Representative VelocityThree alternative equations for the representative velocity were considered in this research: the average velocity in the culvert inlet, the potential flow velocity, and finally the measured flow velocity. Average Flow VelocityThe ABSCOUR program of the MDSHA uses the average velocity in the culvert for the representative velocity.(5) This average velocity, V_{RA}, is just the volumetric flow rate (Q) divided by the cross sectional area of flow in the culvert (A_{CULV}), as in equation 2.
Potential Flow TheoryChang used potential flow principles to derive a velocity adjustment expression to approximate the representative velocity (V_{RP}) that should be used for bridge abutment scour computations.(2) This adjustment compensates for the contraction in flow at the culvert inlet. His expression can be adapted for bottomless culverts, as in equation 3.
where:
Equation 3 applies to a simple contraction, where the unit discharge of the approach section, q_{1}, is less than the unit discharge in the contraction section, q_{2}. The ABSCOUR program states that the values of k_{v} should be limited to the range of values between 1.0 and 1.8.(5) If the computed value is less than 1.0, use a value of 1.0; if the computed value is greater than 1.8, use a value of 1.8. Measured Flow VelocitySince this research produced accurate measurements of the local velocities in the approach section of the culvert, an adjustment was made to the potential flow theory to match the measured flow velocity at the corners of the culvert inlet. This adjustment involved adding a calibration coefficient, C, as given in equation 4.
Critical VelocityThere are two alternatives for calculating the critical velocity at which incipient sediment motion occurs that are considered in this report: Laursen’s method, and Neill’s method. Laursen’s Critical Velocity MethodLaursen’s equation for the critical velocity is summarized in Appendix C of FHWA Hydraulic Engineering Circular No. 18.(6) The critical velocity, V_{CL}, is calculated by equation 5.
where:
Neill’s Competent Velocity MethodNeill presented a family of curves for estimating critical velocities for noncohesive sediments for varying flow depths and with grain sizes ranging from 0.3 to 300 mm (0.0117 to 11.7 inches).(7) Neill defined the critical velocity as the flow velocity just competent to move the bed material. Neill used a combination of field data and laboratory data to develop his family of curves. To develop the family of curves, Neill used a critical velocity equation very similar to Laursen’s to estimate the critical velocity for grain sizes greater than about 30 mm (1.17 inches). For a grain size of 0.3 mm (0.0117 inch), Neill assumed that a regime theory equation for stable channels in sand would be appropriate for estimating the critical velocity. (Regime theory equations are design equations developed from field data collected in the stable, fine sediment canals of Pakistan (Mahmood and Shen)).^{(8)} Having defined critical velocities for a grain size of 0.3 mm (0.0117 inch) and for grain sizes greater than 30 mm (1.17 inches), transition curves were hand drawn for grain sizes between 0.3 and 30 mm (0.0117 and 1.17 inches). Chang transformed the plots of Neill’s curves into a set of equations for computing critical velocity based on the flow depth and the median diameter of the particle.(2) This set is given in equations 6 through 9.
where:
For D_{50} less than 0.03 m (0.1 ft) but greater than 0.0003 m (0.001 ft), Neill’s critical velocity is given in equation 7.
The exponent, x, is calculated using equation 8:
where:
For D_{50} less than 0.0003 m (0.001 ft), Neill’s critical velocity is given in equation 9.
where:
Chang’s equations are plotted in figure 9. Neill’s competent velocity curves are intended for field conditions with flow depths of 1.5 m (5 ft) or greater. Chang’s equations were extrapolated to flow depths below 0.30 m for these experiments and to curves for flow depths of 0.305 and 0.15 m (1 and 0.5 ft) (see figure 9). Note that the sediment sizes used in the experiments fell into the range described by equations 7 and 8.
Adjustment for Spiral Flow at Culvert ToeThis research revealed that the maximum scour depth, y_{max} (measured at the corners of the culvert), was always greater than the computed equilibrium depth, regardless of which equations for representative velocity and critical velocity were used. Thus, an empirical coefficient k_{S}, similar to an adjustment coefficient, was needed to explain the additional scour depth, as in the following equation:
Recalling from the discussion of equation 1 that y_{2} equals q_{R} divided by V_{C} reveals that k_{S} will be a function of V_{R} and V_{C}, among other things. Our research considered two possibilities for a third independent parameter in the equation for k_{S}: the Froude number at the culvert approach, and a dimensionless ratio including Q_{blocked} and y_{2}. Q_{blocked} is the portion of the approach flow that is to one side of the channel centerline and that is blocked by the embankment as the flow approaches the culvert. Equations 11 and 12 give two different functions for k_{S}.
Since there are three different expressions for V_{R}, two different expressions for V_{c}, and two different expressions for the third independent variable, this research considered 12 different k_{S} values. Adjustment for Pressure Flow at a Submerged CulvertThe maximum scour depth, y_{max}, measured under submerged conditions, likewise was always greater than the computed equilibrium depth. Thus, an empirical coefficient, kp, was needed to explain the additional scour depth, as in equation 13.
Equation 14 is the equation for Ak.
where:
Note that due to the influence of k_{S}, this study will also consider 12 different values for k_{p}. Recall also that y_{o} in equation 1 for pressure flow is equal to the hydraulic grade line at the inlet (HGL_{o} in figure 8). These two different adjustment factors will be derived from experimental data for bottomless culverts in the results section. SCOUR PROTECTION: RIPRAP ANALYSISMany researchers have developed critical conditions based on average velocity. Ishbash presented an equation that can be expressed as equation 15.^{(9)}
Ishbash described two critical conditions for riprap stability. For loose stones where no movement occurs, N_{SC} is expressed as equation 16.
For loose stones allowed to roll until they become “seated,” N_{SC} is expressed as equation 17.
where:
Equation 17 for riprap that will just begin to roll can be written as equation 18. For the culvert experiments, we represented the effective velocity (Veff) in terms of an empirical multiplier (equation 19) and the local bed velocity (equation 20), which is substituted into equation 17 to yield equation 21.
where:
Equations 18 through 21 are dimensionally homogeneous and can be used with either system of units as long as they are consistent. 