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Publication Number: FHWA-RD-02-078
Date: November 2003

Bottomless Culvert Scour Study: Phase I Laboratory Report

3. THEORETICAL BACKGROUND

As the photographs in the previous section illustrate, the scour was always deepest near the corners at the upstream entrance to the culvert. This observation is attributed, in part, to the concentration of flow near the upstream corners as the flow that is being blocked by the embankments is contracted and forced to go through the culvert opening. However, it is also attributed to the vortices and strong turbulence generated in the flow separation zone as the blocked flow mixes with the main channel flow at the upstream end of the culvert (figure 10). It is much like the abutment scour phenomenon that researchers have observed for bridge scour models.

Figure 10. Flow concentration and separation zone. Diagram. This drawing depicts the concentration of flow in a current that begins in a wide lane and is forced into a smaller lane that is approximately one-third the width of the original lane. Directly above the point at which the lane narrows, strong backcurrents form, concentrating the flow on the opposite side of the lane.

Figure 10. Flow concentration and separation zone.

Several researchers, including Chang,(1) GKY and Associates, Inc.,(2) and Sturm,(3) 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. The equilibrium flow depth required to balance the sediment load into and out of the scour zone for the assumed flow distribution can be determined analytically. 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 definition sketches (figures 11(a) through 11(c)).

Figure 11A. Definition sketch prior to scour. Diagram. This drawing shows a cross-section of a flume at rest. The bottom of the diagram shows the width of the flume as W subscript A. A model culvert is in the middle of the flume. The culvert is almost as tall as the flume and is approximately one-third the width of the flume, denoted by the measurement W subscript CULV. The original bed is level and is approximately one-fifth the height of the total height of the flume. The water surface in approach is drawn as a straight line across the width of the flume, approximately five-sixths of the flume's height. The water surface in the culvert prior to scour is slightly lower than the water surface in the flume and is denoted by the measurement Y subscript zero.

Figure 11(a). Definition sketch prior to scour.

Figure 11B. Definition sketch after scour. Diagram. This drawing shows a cross-section of a flume after a scour experiment. The bottom of the diagram shows the width of the flume as W subscript A. A model culvert is in the middle of the flume. The culvert is almost as tall as the flume and is approximately one-third the width of the flume, denoted by the measurement W subscript CULV. The original bed is level on the outside of the culvert on both sides and is approximately one-fifth the height of the total height of the flume. The water surface in approach is drawn as a straight line across the width of the flume, approximately five-sixths of the flume's height. The water surface in the culvert prior to scour is slightly lower than the water surface in the flume and is denoted by the measurement Y subscript zero. Inside the culvert, there are measurements that show the actual bed with local scour and a theoretical bed for contraction scour. The actual bed with local scour shown inside the culvert is almost the same height in the center as the bed outside the culvert but slopes downward and is depressed inside near the sides of the culvert. The theoretical contraction scour be is consistently lower than the bed outside the flume, but does not dip as much at the corners of the flume. Y subscript max represents the depth from the culvert's interior water surface to the deepest part of the local scour bed. Y subscript 2 represents the depth from the culvert's interior water surface to the theoretical contraction scour bed. Y subscript max is greater than Y subscript 2, and they are both greater than the original depth Y subscript 0.

Figure 11(b). Definition sketch after scour.

Figure 11C. Definition sketch for blocked area. Diagram. This drawing shows a cross-section of a flume with a blocked area. The bottom of the diagram shows the width of the flume as W subscript A. A model culvert is in the middle of the flume. The culvert is almost as tall as the flume and is approximately one-third the width of the flume, denoted by the measurement W subscript CULV. The original bed is level and is approximately one-fifth the height of the total height of the flume. The bed inside the culvert is almost the same height in the center as the bed outside the culvert but slopes downward and is depressed inside near the sides of the culvert. The water surface in approach is drawn as a straight line across the width of the flume, approximately five-sixths of the flume's height. The water surface in the culvert prior to scour is slightly lower than the water surface in the flume. The blocked area extends through the entire water area outside of the culvert and is denoted by thick, diagonal lines.

Figure 11(c). Definition sketch for blocked area.

Equation 2 is an expression for the unit discharge for an assumed flow distribution remaining 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 velocity must be reduced to the critical incipient motion velocity for the sediment size at the equilibrium flow depth (y2). This equation forms the basis for the analysis:

Equation 2. V subscript R times Y subscript zero, which is the assumed representative unit discharge across the scour hole at the beginning of scour, equals V subscript C, which is the critical incipient motion velocity, times Y subscript 2, which is an equilibrium flow depth.     (2)

where:

VRY0 = qR = the assumed representative unit discharge across the scour hole at the beginning of scour

The above equation can be rearranged to yield an equilibrium flow depth (y2) after the representative velocity (VR) at the beginning of scour and the critical incipient motion velocity (VC) are determined. This equilibrium depth reflects the scour that is attributed to the flow distribution. The measured maximum depth at the corners of the culvert was always greater than the computed equilibrium depth. An empirical coefficient (KADJ), defined by equation 3, was needed to explain the extra scour depths. The laboratory data and regression analyses were used to derive an expression for this coefficient.

Equation 3. K subscript ADJ, which is an empirical coefficient, equals Y subscript max divided by Y subscript 2.     (3)

Several different independent variables were tried to derive the expression for KADJ; however, what seemed to work best for this data was the blocked discharge (Qblocked) normalized by the acceleration of gravity (g) and the computed equilibrium depth (y2) as formulated in equation 4:

Equation 4. K subscript ADJ equals the function of the following quotient: Q subscript blocked, which is the portion of the approach flow to one side of the channel centerline that is blocked by the embankment as the flow approaches the culvert, divided by product of the square root of G, which is the acceleration of gravity, times Y subscript 2 raised to the five-seconds power.     (4)

Qblocked is the portion of the approach flow to one side of the channel centerline that is blocked by the embankment as the flow approaches the culvert.

The literature describes several methods for determining an approximation for representative velocity and critical velocity. Methods described by Chang and by GKY were tried in various combinations to determine which worked best for this data. These methods are discussed below.

CALCULATING REPRESENTATIVE VELOCITY

Maryland DOT (Chang) Method for Representative Velocity

Chang, through his work for the Maryland SHA, developed equations 5 and 6 to calculate the resultant velocity based on potential flow assumptions at a distance equal to one-tenth of the length of the blocked flow (figure 12):

Figure 12. Chang's resultant velocity location. Diagram. This drawing demonstrates potential flow assumptions when nine-tenths of the original flow is blocked. In the diagram, the blockage is shown as a rectangle of length L and extends nine-tenths of the width of the channel, beginning at the left side. The water flow begins at the bottom of the drawing and moves to the right at a representative velocity, V subscript R, denoted by an arrow pointing to the upper right. The width of the remaining opening through which the flow moves is 0.1 L.

Figure 12. Chang's resultant velocity location.

Equation 5. V subscript R equals K subscript V, which is the velocity coefficient to account for flow concentration where side flow converges with main channel flow based on potential flow assumptions, times the quotient of Q, which is the total discharge through the culvert in cubic feet per second, divided by A subscript opening, which is the average flow area within the culvert in square feet.     (5)
Equation 6. K subscript V equals 1 plus 0.8 times the following quotient raised to the 1.5 power: W subscript opening, which is the average flow width in the culvert in feet, divided by W subscript A, which is the width of flow in the approach section in feet.     (6)

where:

KV= velocity coefficient to account for flow concentration where side flow converges with main channel flow based on potential flow assumptions
Q= total discharge through the culvert, ft3/s
Aopening= average flow area within the culvert, ft2
Wopening= average flow width in the culvert, ft
Wa= width of flow in the approach section, ft

These equations are dimensionally homogeneous and are independent of the system of units as long as they are consistent.

GKY Method for Representative Velocity

GKY describes representative velocity across the scour hole as the resultant of the lateral and longitudinal velocity components as shown in equations 7, 8, and 9.

Applying the Pythagorean theorem yields:

Equation 7. V subscript R equals the square root of V subscript X, which is the velocity in the flow direction in feet per second, squared, plus V subscript Y, which is the velocity orthogonal to the flow direction in feet per second, squared.     (7)

with

Equation 8. V subscript X equals Q divided by A subscript opening.     (8)
Equation 9. V subscript Y equals Q subscript blocked, which is the approach flow blocked by the embankment on one side of the channel in CFS, divided by 0.43 times A subscript A, which is the total approach flow area on one side of the channel in square feet.     (9)

It should be noted that equation 9 above is an unpublished modification of the method published by Young, et al.(2); however, the basic concept is similar to the published version.

For the simple rectangular cross section used for the flume experiments, Qblocked could be estimated from equation 10:

Equation 10. Q subscript blocked equals Q times the following quotient: A subscript blocked, which is the approach flow area that is blocked by the embankments on one side of the channel in square feet, divided by A subscript A.     
Vx= velocity in the flow direction, ft/s (figure 13)
Vy= velocity orthogonal to the flow direction, ft/s
Qblocked= approach flow blocked by the embankment on one side of the channel, ft3/s
Aa= total approach flow area on one side of the channel, ft2
Ablocked= approach flow area that is blocked by the embankments on one side of the channel, ft2

Figure 13. GKY's resultant velocity approach. Diagram. This drawing demonstrates visually the approach outlined in equations 7 through 9. The left half of the diagram shows an approach flow area with a solid black line on the left and a dotted line on the right. There is a rectangle, representing a blockage, that extends from the left line to the right line at the top of the drawing, and the area underneath the rectangle is labeled 0.43 A subscript A. The right half of the diagram shows a right-angle triangle, with the base of the triangle as an arrow pointing from left to right labeled V subscript Y, a line on the right extending at a 90-degree angle from the V subscript Y line with an arrow pointing up labeled V subscript X, and the hypotenuse with an arrow touching the top of the V subscript X line that is labeled V subscript R.

Figure 13. GKY's resultant velocity approach.

NUMERICAL MODEL FOR CALCULATING REPRESENTATIVE VELOCITY

Since some designers probably have access to 2D numerical models, they will not necessarily need to rely on the 1D approximations for representative velocity to be used in the computations. Xibing Dou simulated the laboratory experiments with a 2D numerical model. This model uses the Finite Element Surface-Water Modeling System: Two-Dimensional Flow in a Horizontal Plane (FESWMS-2DH) program to solve the hydrodynamic equations that describe 2D flow in the horizontal plane. The effects of bed friction and turbulent stresses are considered and water column pressure is hydrostatic. The estimation of representative velocity uses the average x and y element velocities (velocity x element and velocity y element) for the element transect aligned with the upstream face, excluding the element at the corner and including the next three elements. The numerical model gave rapidly varying velocities in the vicinity of the corners of the culverts. Chang faced a similar problem in interpreting the velocities based on potential flow transformations. Dou tried Chang's selection of a depth-averaged velocity at a distance that was 10 percent of the embankment length into the main channel as illustrated in figure 12, but found better agreement with the 1D approximations by using two locations in a zone that was approximately 10 percent of the culvert width downstream of the culvert face and 25 percent of the culvert width from the culvert wall (figure 14).

Figure 14. Velocity locations for 2D model. Diagram. This is an overhead view of a flume. The flume sidewall is at the top of the drawing, and the mean flow direction travels from the left to the right. A rectangle at the top right of the drawing represents the culvert fill. There is 1 foot between the centerline of the flume and the bottom of the culvert fill rectangle. The first velocity location is directly under the left side of the culvert fill rectangle at a distance of 0.55 feet. The second velocity location is also 0.55 feet below the culvert fill rectangle but is 0.25 feet to the right of the first velocity location.

Figure 14. Velocity locations for 2D model.

Figure 15. Resultant velocity comparison with numerical model at location 2. Graph. This graph compares the two-dimensional model results to the one-dimensional approximations suggested by Chang and GKY. Maximum resultant velocity of numerical model simulation just inside culverts is graphed on the horizontal axis from 0.00 to 1.00 meters per second, and resultant velocity from one-dimensional estimates are graphed on the vertical axis from 0.00 to 1.00 meters per second. The simulated two-dimensional velocity is the local velocity obtained at location 2, as identified in figure 16. A dotted line bisects the graph diagonally from coordinates 0.00, 0.00 to coordinates 1.00, 1.00, representing the line where the 1-D estimate equals the 2-D numerical model simulation's maximum resultant velocity. A text box in the lower right corner of the graph reads, "GKY Estimate, RSQ equals 0.94805, MSE equals 0.00406" and "Chang Estimate, RSQ equals 0.94316, MSE equals 0.00727." Both the GKY and Chang estimates begin near the dotted line at coordinates 0.40, 0.40, and diverge further away from the line as the X and Y coordinates increase, always staying above the dotted line. Thus the 1-D resultant velocity estimate was consistently greater than the 2-D numerical model simultion's resultant velocity.

RSQ = R2 = correlation coefficient
MSE = mean square error
Figure 15. Resultant velocity comparison with numerical model at location 2.

Figure 15 is a comparison of the 2D numerical model results with the 1D approximations suggested by Chang and GKY. The 1D approximations are consistently higher than the numerical results, which is interpreted to mean that the 1D approximations are conservative. Numerical model results could underpredict scour if they are used with empirical equations based on 1D approximations; however, the differences are probably insignificant compared to the differences in the ideal conditions tested in the flume and the conditions that are in a natural channel. In addition to the previous comparison between numerical and 1D measurements, figure 16 shows the comparison between Chang's resultant velocity and GKY's resultant velocity.

Figure 16. Comparison of Chang's and GKY's resultant velocities. Graph. On this graph, the resultant velocity of GKY (V subscript R GKY) is charted on the horizontal axis from 1 to 4, and the resultant velocity of Chang (V subscript R Chang) is charted on the vertical axis from 1 to 4. A solid line bisects the graph diagonally from coordinates 1, 1 to coordinates 4, 4. A text box in the lower right quadrant of the graph reads, "RSQ equals 0.9961." Each data point on the graph represents an experiment in the flume, for which the resultant velocity was calculated by both the Chang and GKY methods and then plotted. The data points hold close to the one-to-one line. That, with the high RSQ value, indicates that the Chang and GKY methods for calculating resultant velocity give similar values.

Figure 16. Comparison of Chang's and GKY's resultant velocities.

CALCULATING CRITICAL VELOCITY

Maryland DOT (Chang) Method for Critical Velocity

Chang uses Niell's(4) competent velocity curves to calculate critical velocity. Niell presents a set of competent (critical) velocity curves based on flow depth, velocity, and the size of the bed material. Niell's curves are derived from the Shields curve using varying Shields numbers for different particle sizes. To facilitate doing computations on a computer spreadsheet, Chang(2) derived a set of simplified equations that represent Niell's curves quite well.

Niell's Competent Velocity Concept

Niell's competent velocity is comparable in its definition to the critical flow velocity for causing incipient motion of bed materials. The equations by Laursen and Niell for computing critical velocity (presented in FHWA's Hydraulic Engineering Circular 18 (HEC-18)) are generally applicable for particles of bed material larger than 0.03 m (0.1 ft). For bed material smaller than this size, these equations can be expected to underestimate critical velocity. Niell developed a series of curves for determining the critical velocity for particles smaller than 0.03 m based on the Shields curve.

Chang transformed the plots of Niell's curves (figure 17) into a set of equations for computing critical velocity based on the flow depth and the median diameter of the particle. These equations are set forth below.

Figure 17. Competent velocity curves for the design of waterway openings in scour backwater conditions (from Neill). Graph. This graph compares four velocity curves at waterway opening depths equal to 5, 10, 20, and 50 feet. Bed material grain size is plotted on the horizontal axis in feet from 0.0010 to 1.000, and competent mean velocity (depth averaged) is plotted on the vertical axis in feet per second from 1.00 to 100.00. All four velocity curves trend generally upward, with velocity increasing as bed material grain size increases, and smaller depths experiencing lower velocities at the same bed material grain size.

Figure 17. Competent velocity curves for the design of waterway openings in scour backwater conditions (from Niell).(4)

  • For D50 > 0.03 m (0.1 ft)
    Equation 11. V subscript C, which is critical velocity, equals the product of K subscript U, which is 1.0 for U.S. customary units and 0.55217 for SI units, times 11.5 times Y subscript 2, which is the equilibrium flow depth in meters or feet, to the one-sixth power, times D subscript 50, which is sediment size in meters or feet, to the one-third power.     (11)

    where:

    y2= equilibrium flow depth, m or ft
    D50= sediment size, m or ft
    KU= 0.55217 for the International System of Units (SI) (metric system) or 1.0 for U.S. customary units


  • For 0.03 m (0.1 ft) > D50 > 0.0003 m (0.001 ft)
    Equation 12. V subscript C equals the product of K subscript U1, which is 1.0 for U.S. customary units and 0.3048 to the 0.65 minus X power, times 11.5 times Y subscript 2 to the X power, times D subscript 50, to the .35 power.     (12)

The exponent x is calculated using equation 13:

Equation 13. The exponent X equals K subscript U2, which is 1.0 for U.S. customary units and 0.788 for SI units, times the quotient of 0.123 divided by D subscript 50 to the 0.20 power.     (13)

where:

y2= equilibrium flow depth, m or ft
D50= sediment size, m or ft
KU1= 0.3048(0.65-x) for SI units or 1.0 for U.S. customary units
X= exponent from equation 13
KU2= 0.788 for SI units or 1.0 for U.S. customary units

  • For 0.0003 m (0.001 ft) > D50
Equation 14. V subscript C equals K subscript U times the square root of Y subscript 2.     (14)

where:

y2= equilibrium flow depth, m or ft
D50= sediment size, m or ft
KU= 0.55217 for SI units or 1.0 for U.S. customary units

Chang's equations are plotted in figure 18. Niell'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) (figure 18). Our sediment sizes fell in a range that could be described by equations 12 and 13.

Figure 18. Chang's approximations. Graph. This graph compares six velocity curves at waterway opening depths equal to 0.5, 1.0, 5, 10, 20, and 50 feet. Sediment size is plotted on the horizontal axis in feet from 0.0010 to 1.000, and competent mean velocity (depth averaged) is plotted on the vertical axis in feet per second from 1.00 to 100.00. All six velocity curves trend generally upward, with velocity increasing as bed material grain size increases, and smaller depths experiencing lower velocities at the same bed material grain size. The velocity curve at a depth of 0.5 feet ends at a sediment size of 0.0500 feet, and the velocity curve at a depth of 1.0 feet ends at a sediment size of 0.1000 feet. The remaining curves extend to the end of the graph.
                                          1 ft = 0.305 m

Figure 18. Chang's approximations.

GKY Method for Critical Velocity

The GKY method combines the Shields,(5) Manning,(6) and Blodgett(7) (SMB) equations to calculate critical velocity. The starting equation is the average shear stress in a control volume of flow:

Equation 15. Tau, the average shear stress in a control volume of flow, equals Gamma times Y subscript 2 times S subscript F.     (15)

Experimental observations highlighted the importance of the Shields parameter (SP), which is defined as:

Equation 16. The Shields parameters (SP) equals tau subscript C divided by the product of the result of Gamma subscript S minus Gamma times D subscript 50.     (16)

The critical value of the stability parameter may be defined at the inception of bed motion, i.e., SP=(SP) c=0.047. Shields showed that (SP)c is primarily a function of the shear Reynolds number (figure 19).

Figure 19. Shields parameter as a function of the particle Reynolds number. Diagram. This diagram shows N subscript R (the particle Reynolds number) as a ray extending from left to right, and SP (the Shields parameter) as a ray extending from the beginning of the N subscript R ray at a 90-degree angle straight up. The critical value of the stability parameter is charted as a dotted line approximately five-sixths of the way to the top of the diagram at 0.047. A curved line begins to the left of the diagram approximately halfway to the top, slopes down as it travels to the right, then slopes upward to slightly cross the dotted line at 0.047 and level out.

Figure 19. Shields parameter as a function of the particle Reynolds number.

Rearranging equation 16, inserting (SP)c = 0.047, and setting Symbol lowercase T from equation 14 equal to Symbol lowercase TC from equation 15 leads to:

Equation 17. 0.047 times the result of Gamma subscript S minus Gamma times D subscript 50 equals Gamma times Y subscript 2 times S subscript F.     (17)

Rearranging Manning's equation to compute the friction slope leads to:

Equation 18. S subscript F equals the product of V squared times N squared divided by the product of K subscript UM, which is 1.49 for U.S. customary units and 1.0 for SI units, squared, times Y subscript 2 to the four-thirds power.     (18)

where:

KUM= 1.0 for SI units or 1.49 for U.S. customary units

Substituting equation 18 for SF in equation 17 results in:

Equation 19. 0.047 times the result of Gamma subscript S minus Gamma times D subscript 50 equals the product of Gamma times Y subscript 2 times V subscript C squared times N squared, divided by the product of K subscript UM squared times Y subscript 2 to the four-thirds power.     (19)

This may be simplified by consolidating the specific weight Symbol lowercase Y and y2 terms:

Equation 20. 0.047 times the result of the quotient of Gamma subscript S divided by Gamma minus 1, times D subscript 50 equals the product of V subscript C squared times N squared, divided by the product of K subscript UM squared times Y subscript 2 to the one-third power.     (20)

Note that:

Equation 21. Gamma subscript S divided by Gamma equals S.G. (specific gravity). (no units)     (21)

Sand such as that used in these experiments is considered to have a specific gravity (SG) of 2.65. Substituting this into equation 20 and rearranging to isolate VC2 leads to:

Equation 22. The product of 0.047 times 1.65 times D subscript 50 times Y subscript 2 to the one-third power times K subscript UM squared, all divided by N squared equals V subscript C squared.     (22)

The square root of equation 22 gives the equation for computing critical velocity:

Equation 23. V subscript C equals the product of K subscript UM times 0.28 times D subscript 50 to the one-half power times Y subscript 2 to the one-sixth power, all divided by N.

Blodgett's equations for average estimates of Manning's n for sand- and gravel-bed channels follow. Equation 25 applies for the depths and sand particle sizes used in our experiments.

Equation 24. N equals the product of K subscript UB, which is 1.0 for U.S. customary units and 1 divided by 1.49 for SI units, times 0.525 times Y subscript 2 to the one-sixth power, all divided by the product of the square root of G, which is the acceleration of gravity, times the sum of 0.794 plus 1.85 log of Y subscript 2 divided by D subscript 50. This is for 1.5 is less than Y subscript 2 divided by D subscript 50 is less than 185.     (24)
Equation 25. N equals the product of K subscript UB times 0.105 times Y subscript 2 to the one-sixth power, all divided by the square root of G. This is for 185 is less than Y subscript 2 divided by D subscript 50 is less than 30,000.     (25)
Where: g= acceleration of gravity
= 9.81 m/s2 for SI units
= 32.2 ft/s2 for U.S. customary units
KUB= 1/1.49 for SI units= 1.0 for U.S. customary units

Substituting equation 25 for n in equation 23 and then simplifying leads to:

Equation 26. V subscript C equals the product of K subscript UB times 0.28 times D subscript 50 to the one-half power times Y subscript 2 to the one-sixth power, all divided by the product of K subscript UB times the quotient of 0.105 Y subscript 2 to the one-sixth power divided by the square root of G.      (26)
Equation 27. V subscript C equals 1.49 times 2.65 D subscript 50 to the one-half power times the square root of G for 185 is less than Y subscript 2 divided by D subscript 50 is less than 30,000.     (27)

Equation 27 is dimensionally homogenous and does not require a units conversion.

Combined Competent Velocity Curves

To give an overview of how the different competent (critical) velocity methods behave, the critical velocity equations for various particle sizes were plotted for 3- and 0.305-m (10- and 1.0-ft) flow depths (figures 20 and 21). Comparing the two plots, the SMB velocity curve drifts away from Chang's approximation and Niell's competent velocity curve for the 3-m (10-ft) flow depth. For a flow depth of 0.305 m (1 ft), this is not the case, which confirms both methods for critical velocity estimation since the experiments were performed in this flow-depth range.

Figure 20. Combined competent velocity curves for a flow depth of 3 meters (10 feet). Graph. This graph compares four velocity equations for various particle sizes at a 10-foot flow depth. The equations plotted are Shields, Manning, Blodgett; Neill's straight line, Neill's competent velocity, and Chang's approximation. Bed material grain size is plotted on the horizontal axis in feet from 0.0010 to 1.000, and competent mean velocity (depth averaged) is plotted on the vertical axis in feet per second from 1.00 to 100.00. All four curves trend generally upward and begin at 0.001-foot grain size, with Sheilds, Manning, Blodgett beginning at 0.7 feet per second, Neill's straight line beginning at 1.8 feet per second, and Neill's competent velocity and Chang's approximation beginning at 3.2 feet per second. The Shields, Manning Blodgett curve remains the lowest of the four; the other three curves converge at coordinates 0.100, 8.00 and continue upward on identical paths.
                                              1 ft = 0.305 m

Figure 20. Combined competent velocity curves for a flow depth of 3 m (10 ft).

Figure 21. Combined competent velocity curves for a flow depth of 0.3 meters (1 foot). Graph. This graph compares four velocity equations for various particle sizes at a 1-foot flow depth. The equations plotted are Shields, Manning, Blodgett; Neill's straight line, Neill's competent velocity, and Chang's approximation. Bed material grain size is plotted on the horizontal axis in feet from 0.0010 to 1.000, and competent mean velocity (depth averaged) is plotted on the vertical axis in feet per second from 1.00 to 100.00. All four curves trend generally upward and begin at 0.001-foot grain size, with Sheilds, Manning, Blodgett beginning at 0.7 feet per second, Neill's straight line and Chang's approximation beginning at 1.00 feet per second, and Neill's competent velocity beginning at 1.02 feet per second. In general, the curves follow parallel paths.
                                            1 ft = 0.305 m

Figure 21. Combined competent velocity curves for a flow depth of 0.3 m (1.0 ft).

SCOUR PROTECTION TASK: RIPRAP ANALYSIS

Many researchers have developed critical conditions based on average velocity. Ishbash(8) presented an equation that can be expressed as:

Equation 28. N subscript SC, which is the computed sediment number for distributed flow, equals E, which is a constant.     (28)

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

Equation 29. N subscript SC equals V subscript min, which is the minimum velocity that will remove the loose stones lying on top of the fill in feet per second, squared, divided by the product of 2 times G, which is the acceleration of gravity, times D subscript 50, which is the diameter of riprap, times the result of SG, which is the specific gravity of riprap, minus 1. E equals 0.86.     (29)

For loose stones allowed to roll until they become "seated", NSC is expressed as:

Equation 30. N subscript SC equals V subscript max, which is the maximum velocity that will remove the loose stones lying on top of the fill in feet per second, squared, divided by the product of 2 times G times D subscript 50 times the result of SG minus 1. E equals 1.44.     (30)

where:

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

Equation 30 for riprap that will just begin to roll can be written as equation 31. For the culvert experiments, we represented the effective velocity (Veff) in terms of an empirical multiplier as indicated by equation 32, which is substituted into equation 31 to yield equation 33.

Equation 31. D subscript 50 equals 0.69 times V subscript eff, which is the effective velocity that accounts for turbulence and vorticity in the mixing zone at the upstream corner of a culvert, squared, divided by the product of 2 times G times the result of SG minus 1.     (31)
Equation 32. V subscript eff equals K subscript RIP times V subscript R, which is the presumed representative velocity prior to scour in the vicinity of the upstream corner of a culvert.     (32)
Equation 33. K subscript RIP equals 1.20 times the square root of the product of 2 times G times the result of SG minus 1 times D subscript 50, all divided by V subscript R.     (33)

where:

Veff= effective velocity that accounts for turbulence and vorticity in the mixing zone at the upstream corner of a culvert
VR= presumed representative velocity prior to scour in the vicinity of the upstream corner of a culvert
D50= diameter of riprap that is expected to be on the verge of failure in the vicinity of the upstream corner of the culvert

Equations 31 through 33 are dimensionally homogeneous and can be used with either system of units as long as they are consistent.

Regression analysis was then performed to derive a function for the coefficient KRIP.


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The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). The hydraulics and hydrology research program at the TFHRC Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
FHWA
United States Department of Transportation - Federal Highway Administration