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This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-HRT-07-026
Date: February 2007

Bottomless Culvert Scour Study: Phase II Laboratory Report

Chapter 3: Theoretical Background

Experiments 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 so-called separation zone. This flow pattern is very similar to the abutment scour phenomenon that researchers have observed for bridge scour.

Figure 4. Diagram. Flow concentration and separation zone. The two-dimensional diagram shows the flow into a culvert. The left side of the diagram is a rectangular approach area to the culvert, which is depicted as a smaller rectangle abutting the center of the approach area. The primary flow, indicated by arrows, enters the approach area, curves into the culvert entrance, narrows in the initial portion of the culvert, and then expands to fill the culvert. In the initial portion of the culvert where the primary flow narrows, secondary flows, or eddies, occur, The secondary flows are indicated by pinwheels on the underside of the top edge of the culvert and the upper side of the bottom edge of the culvert.
Figure 4. Diagram. Flow concentration and separation zone.

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.

Figure 5. Diagram. Definition sketch before scour for unsubmerged flow conditions. The two-dimensional diagram shows water levels in a culvert prior to scour. The view is into the culvert entrance, which is in the center of the diagram. Each side of the culvert entrance is a straight vertical line. The top of the culvert entrance is formed by an extension of the side lines to join and form a semi-ellipse facing downward. The width of the culvert entrance is labeled w subscript C U L V. The approach section to the culvert entrance extends to the left and right on each side of the entrance. The width of the approach section is labeled w subscript a. The water surface in the approach section is indicated by a horizontal line that intersects the culvert entrance near the entrance’s top. The depth of the water in the approach section is the vertical distance, labeled y subscript 1, between the water surface and the original bed of the approach section. The water surface in the culvert prior to scour is lower than the water surface in the approach section. The distance from the water surface in the culvert to the bed in the culvert, which is the same elevation as the original bed, is labeled y subscript o.
Figure 5. Diagram. Definition sketch before scour for unsubmerged flow conditions.

Figure 6. Diagram. Definition sketch after scour for unsubmerged flow conditions. The two-dimensional diagram shows water levels in a culvert after scour. The view is into the culvert entrance, which is in the center of the diagram. Each side of the culvert entrance is a straight vertical line. The top of the culvert entrance is formed by an extension of the side lines to join and form a semi-ellipse facing downward. The width of the culvert entrance is labeled w subscript C U L V. The approach section to the culvert entrance extends to the left and right on each side of the entrance. The width of the approach section is labeled w subscript a. The water surface in the approach section is indicated by a horizontal line that intersects the culvert entrance near the entrance’s top. Three beds are indicated in the diagram. A horizontal line to the left and right of the culvert entrance, and a short distance above the bottom of the culvert entrance, is the original bed. The other two beds are inside the culvert entrance. The theoretical bed for contraction scour is indicated by a horizontal line that is a short distance above the bottom of the culvert entrance but below the level of the original bed line. The actual bed with local scour is the other bed inside the culvert entrance. At each end, the actual bed with local scour is above the bottom of the culvert entrance but below the horizontal line indicating the theoretical bed for contraction scour. In the approximate center two-thirds of the culvert entrance, the actual bed with local scour is above the theoretical bed for contraction scour. The water surface in the culvert prior to scour is lower than the water surface in the approach section. The distance between the water surface in the approach section and the original bed is labeled y subscript 1. The distance from the water surface in the culvert to the original bed is labeled y subscript o. The distance from the water surface in the culvert to the theoretical bed for contraction scour is labeled y subscript max. The distance from the water surface in the culvert to the highest point of the actual bed with local scour is labeled y subscript 2. The distance between the original bed and the theoretical bed for contraction scour is labeled y subscript s.
Figure 6. Diagram. Definition sketch after scour for unsubmerged flow conditions.

Figure 7. Diagram. Definition sketch after scour for submerged flow conditions. The two-dimensional diagram shows water levels in a culvert after scour for submerged flow conditions. The view is into the culvert entrance, which is in the center of the diagram. Each side of the culvert entrance is a straight vertical line. The top of the culvert entrance is formed by an extension of the side lines to join and form a semi-ellipse facing downward. The width of the culvert’s entrance is labeled w subscript C U L V. The approach section to the culvert entrance extends to the left and right on each side of the entrance. The width of the approach section is labeled w subscript a. The water surface in the approach section is indicated by a horizontal line that is above the culvert entrance. Three beds are indicated in the diagram. A horizontal line to the left and right of the culvert entrance, and a short distance above the bottom of the culvert entrance, is the original bed. The other two beds are inside the culvert entrance. The theoretical bed for contraction scour is indicated by a horizontal line that is a short distance above the bottom of the culvert entrance but below the level of the original bed. The other bed inside the culvert entrance is the actual bed with local scour. At each end, the actual bed with local scour is above the bottom of the culvert entrance but below the horizontal line indicating the theoretical bed for contraction scour. In the approximate center two-thirds of the culvert entrance, the actual bed with local scour is above the theoretical bed for contraction scour. The distance between the water surface in the approach section and the original bed is labeled y subscript 1. The distance between the original bed and the theoretical bed for contraction scour is labeled y subscript s. A vertical line labeled A at the top and A prime at the bottom extends the height of the diagram at the diagram’s center. The vertical line marks a cross section that is the subject of figure 8.
Figure 7. Definition sketch after scour for submerged flow conditions

Figure 8. Diagram. Side view after scour for submerged flow conditions, Section A-A prime in figure 7. This figure gives a side view of the culvert entrance depicted in figures 5, 6, and 7. The flow is from right to left, the culvert entrance is in the center of the figure, and the culvert barrel is to the left. The Energy Grade Line, or E G L, is at the top of the figure. The E G L is horizontal in the right portion of the figure. At a point directly above the culvert entrance, the E G L slopes downward to the left. Just below the E G L on the right side of the figure is a horizontal line indicating the water surface. The line stops directly above the culvert entrance. The distance between the water surface and the bed in the approach section to the culvert entrance is labeled y subscript 1.The Hydraulic Grade Line, or H G L, begins above the culvert entrance at a point below the water surface line and slopes downward to the left, parallel to the E G L. A horizontal line extends to the left and the right from the point where the H G L intersects a vertical line above the culvert entrance. A concave depression facing upwards at the culvert entrance depicts the effects of scour in the culvert bed at the entrance. The distance from the H G L to the culvert bed before scouring is labeled H G L subscript o equals y subscript o. The distance from the point where the H G L intersects a vertical line above the culvert entrance and the culvert bed after scouring is labeled y subscript 2. The distance between the culvert bed before scouring and the culvert bed after scouring is labeled y subscript s. The distance between the water surface in the approach section and the original bed is labeled y subscript 1.
Figure 8. Diagram. Side view after scour for submerged flow conditions (Section A-A′ in figure 7).

wCULV
is width of the culvert.
wa
is width of the approach channel.
y1
is water depth in the approach channel at a distance three times wCULV upstream of the culvert entrance.
y0
is water depth at the culvert entrance before scour occurs.
ymax
is maximum water depth in the culvert after scour hole develops.
y2
is equilibrium water depth after scour hole develops.
yS
is maximum depth of scour in the culvert.

CLEAR WATER SCOUR

Equation 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, Vc, for the sediment size at the equilibrium flow depth, y2. This equation forms the basis for the analysis:

1. The product of V subscript R times y subscript 0 equals the product of V subscript C times y subscript 2. (1)

where:

VR
is representative (local) velocity at the entrance of the culvert.
VC
is critical velocity at which incipient sediment motion occurs.

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 qR.

Equation 1 can be rearranged to yield an equilibrium flow depth, y2, once the representative velocity, VR, and the critical incipient motion velocity, VC, 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 Velocity

Three 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 Velocity

The ABSCOUR program of the MDSHA uses the average velocity in the culvert for the representative velocity.(5) This average velocity, VRA, is just the volumetric flow rate (Q) divided by the cross sectional area of flow in the culvert (ACULV), as in equation 2.

2. V subscript R A equals the quotient of uppercase Q divided by uppercase A subscript C U L V. This quotient in turn equals the quotient of uppercase Q divided by the product of y subscript 0 times w subscript C U L V. (2)
Potential Flow Theory

Chang used potential flow principles to derive a velocity adjustment expression to approximate the representative velocity (VRP) 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.

3. V subscript R P equals the product of lowercase k subscript V times V subscript R A. This product in turn equals the product of two terms. The first term is the sum of one plus the product of 0.8 times the quotient to the 1.5 power of lowercase q subscript 1 divided by lowercase q subscript 2. The second term is the quotient of uppercase Q divided by the product of y subscript 0 times w subscript C U L V. (3)

where:

kV
is the ratio of velocity at the culvert toe to the mean velocity in the contracted section.
q1
is unit discharge in the approach section.
q2
is unit discharge in the contracted section.

Equation 3 applies to a simple contraction, where the unit discharge of the approach section, q1, is less than the unit discharge in the contraction section, q2. The ABSCOUR program states that the values of kv 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 Velocity

Since 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.

4. V subscript R M equals the product of lowercase k subscript V a d j times V subscript R A. This product in turn equals the product of three terms. The first term is C. The second term is the sum of one plus the product of 0.8 times the quotient to the 1.5 power of lowercase q subscript 1 divided by lowercase q subscript 2. The third term is the quotient of uppercase Q divided by the product of y subscript 0 times w subscript C U L V. (4)
Critical Velocity

There 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 Method

Laursen’s equation for the critical velocity is summarized in Appendix C of FHWA Hydraulic Engineering Circular No. 18.(6) The critical velocity, VCL, is calculated by equation 5.

5. V subscript C L equals the product of uppercase K subscript u times the one-sixth power of y subscript 2 times the one-third power of D subscript 50. (5)

where:

Ku
is 6.19 for SI units, or 11.17 for U.S. customary units.
y2
is equilibrium scour flow depth (m or ft).
D50
is sediment size (m or ft).
Neill’s Competent Velocity Method

Neill 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.
For D50 greater than 0.03 m (0.1 ft), Neill’s critical velocity, VCN, is given in equation 6.

6. V subscript C N equals the product of uppercase K subscript U times 11.5 times the one-sixth power of y subscript 2 times the one-third power of D subscript 50. (6)

where:

y2
is equilibrium scour flow depth (m or ft).
D50
is sediment size (m or ft).
Ku
is 0.55217 for SI units, or 1.0 for U.S. customary units.

For D50 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.

7. V subscript C N equals the product of uppercase K subscript U1 times 11.5 times the x power of y subscript 2 times the 0.35 power of D subscript 50. (7)

The exponent, x, is calculated using equation 8:

8. Lowercase x equals the product of uppercase K subscript U2 times the quotient of 0.123 divided by the 0.20 power of D subscript 50. (8)

where:

y2
is equilibrium flow depth (m or ft).
D50
is sediment size (m or ft).
KU1
is, for SI units, 0.3048 to the power of 0.65 minus x, or 1.0 for U.S. customary units.
x
is the exponent as calculated in equation 8.
KU2
is 0.788 for SI units, or 1.0 for U.S. customary units.

For D50 less than 0.0003 m (0.001 ft), Neill’s critical velocity is given in equation 9.

9. V subscript C N equals the product of uppercase K subscript U times the square root of y subscript 2. (9)

where:

y2
is equilibrium flow depth (m or ft).
D50
is sediment size (m or ft).
Ku
is 0.55217 for SI units, or 1.0 for U.S. customary units.

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.

Figure 9. Graph. Chang’s approximations to Neill’s competent velocity curves. The graph contains plots for five depths in feet: 0.5, 1.0, 5, 10, 20, and 50. The y-axis is the critical velocity at which incipient sediment motion occurs, V subscript c, and ranges from 0.10 to 100.0 feet per second. The x-axis is D subscript 50, which is the median grain size of bed material, and ranges from 0.0010 to 1.0000 feet. Both the y-axis and x-axis are logarithmic scales. Each plot has an overall rise from left to right. The lowest plot is for a depth of 0.5 feet. Its leftmost point has x-axis and y-axis coordinates of approximately 0.0010 and 0.70, respectively. Its rightmost point has x-axis and y-axis coordinates of approximately 0.0140 and 1.25, respectively. The highest plot on the graph is for a depth of 50 feet. Its leftmost point has x-axis and y-axis coordinates of approximately 0.0010 and 1.75, respectively. Its rightmost point has x-axis and y-axis coordinates of approximately 1.0000 and 10.12, respectively. One foot equals 0.305 meters or 305 millimeters.
Figure 9. Graph. Chang’s approximations to Neill’s competent velocity curves.

Adjustment for Spiral Flow at Culvert Toe

This research revealed that the maximum scour depth, ymax (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 kS, similar to an adjustment coefficient, was needed to explain the additional scour depth, as in the following equation:

10. The quotient of y subscript max divided by y subscript 2 equals lowercase k subscript s. (10)

Recalling from the discussion of equation 1 that y2 equals qR divided by VC reveals that kS will be a function of VR and VC, among other things. Our research considered two possibilities for a third independent parameter in the equation for kS: the Froude number at the culvert approach, and a dimensionless ratio including Qblocked and y2. Qblocked 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 kS.

11. Lowercase k subscript s is a function, which is identified as lowercase f subscript 1, of V subscript R, V subscript C, and uppercase F subscript 1. Lowercase F subscript 1 is the quotient of uppercase Q divided by the product of y subscript 1 times w subscript a times the square root of the product of g times y subscript 1. (11)
12. Lowercase k subscript s is a function, which is identified as lowercase f subscript 2, of V subscript R, V subscript C, and R subscript uppercase Q blocked. R subscript uppercase Q blocked is the quotient of uppercase Q subscript blocked divided by the product of the square root of g times the five-seconds power of y subscript 2. (12)

Since there are three different expressions for VR, two different expressions for Vc, and two different expressions for the third independent variable, this research considered 12 different kS values.

Adjustment for Pressure Flow at a Submerged Culvert

The maximum scour depth, ymax, 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.

13. The quotient of y subscript max divided by y subscript 2 equals the product of lowercase k subscript p times lowercase k subscript s. This product in turn is a product of two terms. The first term is a function, which is identified as lowercase f subscript 3, of uppercase A subscript lowercase k. The second term is one of two functions. The first possible function, which is identified as lowercase f subscript 1, is of V subscript R, V subscript C, and uppercase F subscript 1. The second possible function, which is identified as lowercase f subscript 2, is of V subscript R, V subscript C, and R subscript Q blocked. (13)

Equation 14 is the equation for Ak.

14. Uppercase A subscript lowercase k equals a quotient. The numerator of the quotient is w subscript C U L V times the difference of y subscript 1 minus D. The denominator of the quotient is the product of w subscript lowercase a times y subscript 1. (14)

where:

D
is the culvert height at the approach prior to scour.
Ak
is a dimensionless ratio: area of approaching flow directly above the culvert divided by the total area of flow approaching the culvert.

Note that due to the influence of kS, this study will also consider 12 different values for kp. Recall also that yo in equation 1 for pressure flow is equal to the hydraulic grade line at the inlet (HGLo in figure 8). These two different adjustment factors will be derived from experimental data for bottomless culverts in the results section.

SCOUR PROTECTION: RIPRAP ANALYSIS

Many researchers have developed critical conditions based on average velocity. Ishbash presented an equation that can be expressed as equation 15.(9)

15. N subscript S C equals E. (15)

Ishbash described two critical conditions for riprap stability. For loose stones where no movement occurs, NSC is expressed as equation 16.

16. N subscript S C equals the quotient of the square of V subscript min divided by the product of 2 times lowercase g times D subscript 50 times the difference of S G minus 1. (16)

For loose stones allowed to roll until they become “seated,” NSC is expressed as equation 17.

17. N subscript S C equals the quotient of the square of V subscript max divided by the product of 2 times g times D subscript 50 times the difference of SG minus 1. (17)

where:

NSC
is computed sediment number for distributed flow.
Vmin
is minimum velocity (ft/s) that will remove the loose stones lying on top of the fill.
Vmax
is maximum velocity (ft/s) that will roll out the stones lying among the others on the slope.
g
is acceleration of gravity (ft/s2).
D50
is diameter of riprap (ft).
SG
is specific gravity of riprap.
E
is the Ishbash constant.

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.

18. D subscript 50 equals 0.69 times the quotient of the square of V subscript e f f divided by the product of 2 times lowercase g times the difference of S G minus 1. (18)
19. V subscript e f f equals the product of uppercase K subscript R I P times V subscript L B. (19)
20. V subscript L B equals the product of uppercase K subscript V M times V subscript A C. (20)
21. Uppercase K subscript R I P equals a quotient. The numerator of the quotient is 1.20 times the square root of a product. The product is 2 times lowercase g times D subscript 50 times the difference of S G minus 1. The denominator of the quotient is V subscript L B. (21)

where:

Veff
is effective velocity that accounts for turbulence and vorticity in the mixing zone at the upstream corner of a culvert.
VLB
is local velocity along the bed prior to scour in the vicinity of the upstream corner of a culvert.
VAC
is average velocity in the contracted zone prior to scour in the vicinity of the upstream corner of a culvert.
KRIP
is the coefficient used to size riprap for scour (to be determined in lab experiments).
KVM
is the coefficient relating the local bed velocity in the experiments to the average velocity in the contraction zone (to be determined in lab experiments).
D50
is the diameter of riprap that is expected to be on the verge of failure in the vicinity of the upstream corner of the culvert.

Equations 18 through 21 are dimensionally homogeneous and can be used with either system of units as long as they are consistent.

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