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Publication Number:  FHWA-HRT-17-013     Date:  February 2017
Publication Number: FHWA-HRT-17-013
Date: February 2017

 

Hydraulic Performance of Shallow Foundations for The Support of Vertical-Wall Bridge Abutments

CHAPTER 3. CONCEPTUAL MODEL FOR FLUSH RIPRAP APRONS

One hypothesis for this research effort was that the use of riprap aprons installed flush with the bed in some bridge openings might increase the depth of contraction scour within the opening between the riprap aprons. In this chapter, a conceptual model illustrating the basis for this hypothesis is developed.

INCREASED STRESS FROM THE RIPRAP APRON

Figure 2 shows a simplified plan view of a river channel with bridge abutments encroaching into the channel. Riprap aprons are also shown as a protection against abutment scour. The hypothesis was that the presence of these aprons for abutment scour had the potential to increase contraction scour in the area of the channel in the unprotected gap between the aprons. The relevant hydraulic parameters are also shown in the figure, including the upstream velocity (V1,), velocity in the abutment opening (V2), riverbed bottom width upstream (W1), the bridge opening (W2), the flow depth upstream (y1), abutment length (l1), and abutment width (l2).

This figure shows the upstream velocity, V sub 1, upstream depth, y sub 1, and upstream width, W sub 1, along with the opening flow depth before scour, y sub 0, the opening velocity, V sub 2, and the opening width, W sub 2. The figure shows the placement of the abutment with defining the length, l sub 1 and width, l sub 2. The figure also defines the riprap apron extent, W sub R.
WR = Riprap apron width.
Figure 2. Sketch. Plan view of a bridge contraction with riprap apron.

 

Figure 3 shows a schematic of a simple bridge opening (contraction) without a riprap apron. As previously defined, W1 is the bottom width of the upstream (approach) channel, and W2 is the bottom width of the contracted (bridge) opening. The average contraction scour in the bridge opening is yc.

This study focused on vertical-wall abutments, as shown in figure 3. These abutments take many forms, including GRS abutments placed on RSFs. This type of abutment and foundation is considered a non-rigid foundation. According to design guidance in HEC-23, the top of the abutment foundation should be located at the lower extent of the estimated contraction scour and LTD.(3) LTD was negligible in the physical modeling because equilibrium conditions were used.

This figure illustrates a simple bridge opening cross section with vertical wall abutments supported by shallow foundations on both sides. The upstream width, W sub 1, and opening width, W sub 2, are indicated, along with the depth before scour, y sub 0, and the contraction scour depth, y sub c.
Figure 3. Sketch. Cross section of pre-scour condition for case without riprap.

 

Figure 4 shows the contracted bridge opening with the riprap apron to protect the abutment foundation from abutment scour as specified in HEC-23. (3)The apron width, WR, is extended from the abutment face a distance equal to twice the pre-scour flow depth in the opening, y0. The recommended apron thickness, according to HEC-23, is 1.5×D50 or D100 (the maximum rock size), whichever is greater. It is shown in the figure from the pre-scour bed elevation to the top of the foundation.(23)

This figure shows the same information as figure 3 with a riprap apron and filter fabric added. The riprap apron is shown extending a distance W sub r from the abutment wall, which is equal to 2 times y sub 0. The apron thickness is shown equal to the contraction scour depth.
Figure 4. Sketch. Cross section of pre-scour condition for case with riprap.

 

When the contraction scour for the situation shown in figure 3 occurred, a shear stress remained on the bed material. If this was the equilibrium contraction scour depth, the shear stress would equal the critical shear stress for the bed material. To generalize, that shear stress on the bed without riprap aprons was referred to in this study as τB. Similarly, when the same contraction scour shown in figure 4 occurred but with riprap aprons present, that shear stress was defined in this study as τR. Although both shear stresses were considered at the same depth (y0 + yc), the hypothesis of this study was that the presence of the riprap caused τR to exceed τB so that further contraction scour would occur for the case with riprap.

Numerical CFD modeling, which is discussed in chapter 5, supported this hypothesis. Figure 5 shows the bed shear stress distribution for a case without riprap after contraction scour occurred. In the region of interest, the value of τB was approximately 0.023 lbf/ft2 (1.1 Pa). Figure 6 displays the bed shear stress distribution at the same level of scour for the same contraction geometry, but with riprap aprons. In the same region of interest, the shear stress was higher, such that τR was approximately 0.031 lbf/ft2 (1.5 Pa). The difference between the two values depended on the specific circumstances of the contraction and riprap characteristics. The shear ratio τR/τB was used as an index to evaluate the influence induced by a riprap installation.

This figure is a computer-generated graphic showing shear stress over the area in the contraction and upstream and downstream of the contraction. The figure shows a rectangular area in the center of the contracted section where the quantity tau sub B is measured. The figure shows elevated shear stresses at the upstream edges of the abutment corners and to a lesser extent elsewhere in the contracted section. (1 lbf/ft<sup>2</sup> = 47.88 Pa)
1 lbf/ft2 = 47.88 Pa.
Figure 5. Graphic. Numerically modeled shear stress distribution for case without riprap.

 

This figure is a computer-generated graphic showing shear stress over the area in the contraction and upstream and downstream of the contraction and also showing the extent of the riprap apron. The figure shows a rectangular area in the center of the contracted section where the quantity tau sub R is measured. The elevated shear stresses are in the center of the contracted area between the edges of the riprap apron placed around the edges of the abutment on both sides of the contracted section. (1 lbf/ft<sup>2</sup> = 47.88 Pa)
1 lbf/ft2 = 47.88 Pa.
Figure 6. Graphic. Numerically modeled shear stress distribution for case with riprap.

 

DIMENSIONAL ANALYSIS

A dimensional analysis was used to better understand the relationship between the ratio of shear with and without riprap abutment protection to potential independent variables. The general equations for the case without riprap and with riprap are shown in figure 7 and figure 8, respectively.

The function, f sub 1, of open parenthesis V sub 1, V sub 2, rho, g, l sub 1, l sub 2, tau sub B, mu, W sub 1, W sub 2, y sub 1, y sub 2 close parenthesis equals 0.

Figure 7. Equation. Relevant variables for case without riprap.

 

The function, f sub 2, of open parenthesis V sub 1, V sub 2, rho, g, l sub 1, l sub 2, D sub 50, tau sub R, mu, W sub 1, W sub 2, y sub 1, y sub 2 close parenthesis equals 0.

Figure 8. Equation. Relevant variables for case with riprap.

 

Where:

ρ = Water density (lb/ft3 (kg/m3)).
g = Acceleration resulting from gravity (ft/s2 (m/s2)).
μ = Viscosity (lbf∙s/ft2 (N s/m2)).

These equations can be expressed as ratios, normalizing using the upstream hydraulic variables and water density, as shown in figure 9 and figure 10.

The function, f sub 1, of open parenthesis V sub 2 divided by V sub 1, g times y sub 1 divided by V sub 1 squared, l sub 1 divided by y sub 1, l sub 2 divided by y sub 1, tau sub B divided by the quantity V sub 1 squared times rho times by y sub 1 end quantity, mu divided by the quantity V sub 1 times rho times y sub 1, W sub 1 divided by y sub 1, W sub 2 divided by y sub 1, y sub 2 divided by y sub 1 close parenthesis equals 0.

Figure 9. Equation. Normalized variables for case without riprap.

 

The function, f sub 2, of open parenthesis V sub 2 divided by V sub 1, g times y sub 1 divided by V sub 1 squared, l sub 1 divided by y sub 1, l sub 2 divided by y sub 1, D sub 50 divided by y sub 1, tau sub R divided by the quantity V sub 1 squared times rho times y sub 1, mu divided by the quantity V sub 1 times rho times y sub 1 end quantity, W sub 1 divided by y sub 1, W sub 2 divided by 
y sub 1, y sub 2 divided by y sub 1 close parenthesis equals 0.

Figure 10. Equation. Normalized variables for case with riprap.

 

These relations can be reduced further by recognizing the following:

With these observations, the cases without and with riprap are summarized in figure 11 and figure 12, respectively.

The function, f sub 1, of open parenthesis g times y sub 1 divided by V sub 1 squared, l sub 1 divided by y sub 1, tau sub B divided by the quantity V sub 1 squared times rho times y sub 1 end quantity, mu divided by the quantity V sub 1 times rho times y sub 1, W sub 2 divided by y sub 1 close parenthesis equals 0.

Figure 11. Equation. Reduced function for case without riprap.

 

The function, f sub 2, of open parenthesis g times y sub 1 divided by V sub 1 squared, l sub 1 divided by y sub 1, tau sub R divided by the quantity V sub 1 squared times rho times y sub 1, mu divided by the quantity V sub 1 times rho times y sub 1, W sub 2 divided by y sub 1 close parenthesis equals 0.

Figure 12. Equation. Reduced function for case with riprap.

 

Conversion of these equations to a formulation expressing the shear is shown in figure 13 and figure 14 for the cases without and with riprap, respectively.

tau sub B divided by the quantity V sub 1 squared times rho times y sub 1 end quantity equals the function, f sub 1, of open parenthesis Fr, Re, l sub 1 divided by y sub 1, W sub 2 divided by y sub 1 close parenthesis.

Figure 13. Equation. Shear function for case without riprap.

 

tau sub R divided by the quantity  V sub 1 squared times rho times y sub 1 end quantity equals the function, f sub 2, of open parenthesis Fr, Re, l sub 1 divided by y sub 1, W sub 2 divided by y sub 1 close parenthesis.

Figure 14. Equation. Shear function for case with riprap.

 

Where:

Fr = Froude number (dimensionless).
Re = Reynolds number (dimensionless).

Then, the ratio of shear stress with riprap to the case without riprap can be written as shown in figure 15.

tau sub R divided tau sub B equals the function, f sub 3, of open parenthesis Fr, Re, l sub 1 divided by y sub 1, W sub 2 divided by y sub 1 close parenthesis.

Figure 15. Equation. Shear ratio of case with riprap to case without riprap.

 

The Reynolds number (Re) is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces. For this application, Re was not important because the bed shear primarily depended on the local velocity gradient on the bed.

Further, if a constant abutment length was assumed, the term with l1 was not needed. Dropping the terms with Re and l1 yielded the formulation shown in figure 16.

tau sub R divided tau sub B equals the function phi of open parenthesis Fr, W sub 2 divided by y sub 1 close parenthesis.

Figure 16. Equation. Final shear ratio of case with riprap to case without riprap.

 

Where:

∅ = Undefined function.

DERIVATION OF THE FUNCTIONAL FORM

The specific functional formulation expressing the ratio of τR / τB was derived from the continuity, energy, and momentum equations comparing the situations with and without riprap aprons.

Natural Bed Materials: No Riprap

Figure 17 provides a conceptual definition sketch of the approach (upstream) section and contracted (bridge opening) section for the case where there is no riprap. To simplify the numerical modeling with CFD, an alternative definition sketch was used, as shown in figure 18. The change from the conceptual definition was to alter the upstream section so that the bottom elevation was increased by the estimated contraction scour depth, yc.

This figure shows a side view of the approach flow with a depth of y sub 1 and illustrates the variables at the abutment, including y sub 0, y sub 2, and y sub c. A front view shows the abutments on both sides, with the riverbed before scour at a depth of y sub 0 and the contraction scour y sub c. The opening width, W sub 2, between the abutments is shown.
Figure 17. Sketch. Conceptual schematic for case without riprap.

 

This figure shows a side view of the approach flow with a depth of y sub 1 plus y sub c and illustrates the variables at the abutment including y sub 0, y sub 2, and y sub c. A front view shows the abutments on both sides, with the riverbed before scour at a depth of y sub 0 and the contraction scour y sub c. The opening width, W sub 2, between the abutments is shown.
Figure 18. Sketch. CFD schematic for case without riprap.

 

The flow condition in the opening without riprap at equilibrium contraction scour is shown in figure 19. For this case, the bottom roughness for the natural bed material is nB and the average shear stress on the bed is τB.

This figure shows a rectangle with depth of y sub 2 and width of W sub 2. The area is A sub 2. The roughness, n sub B, and shear, tau sub B, are shown.
Figure 19. Schematic. Cross section in the opening for case without riprap.

 

Figure 20 displays a schematic consistent with the CFD analysis for use with the continuity and energy equations. The variable ŷ1 is equal to y1 + yc, as shown in figure 18.

This figure shows an isometric control volume with the approach (section 1) and the contraction (section 2). The variables shown in section 1 are V hat sub 1, A hat sub 1, and y hat sub 1. The variables shown in section 2 are V sub 2, A sub 2, and y sub 2.
velosity1= Average velocity in upstream section for CFD experiments.
Â1 = Cross-sectional area of upstream section for CFD experiments.
ŷ1 = Average flow depth in upstream section for CFD experiments.
y2 = Average flow depth in contracted section with riprap after contraction scour.
A2 = Cross-sectional area of contracted section without riprap at equilibrium contraction scour.
Figure 20. Schematic. Continuity and energy control volume for case without riprap.

 

For the case without riprap, the continuity equation is summarized in figure 21, and the energy equation is summarized in figure 22.

V hat sub 1 times A hat sub 1 equals V sub 2 times A sub 2.

Figure 21. Equation. Continuity equation for case without riprap.

 

z sub 1 plus y hat sub 1 plus alpha sub 1 times V hat sub 1 squared divided by 2g equals z sub 2 plus y sub 2 plus the quantity alpha sub 2B times v sub 2 squared divided by 2g end quantity plus h sub B.

Figure 22. Equation. Energy equation for case without riprap.

 

Where:

z1 = Reference elevation for upstream section (ft (m)).
z2 = Reference elevation for downstream section (ft (m)).
α1 = Energy correction factor for upstream section (dimensionless).
α2B = Energy correction factor for contracted section without riprap (dimensionless).
hB = Head loss between upstream section and contracted section without riprap (ft (m)).

The energy correction factors were approximately equal to 1.00 to 1.01 for gradually varied flow. The head loss between to the two sections without riprap, hB, is defined as the contraction coefficient without riprap, CcB, times the velocity head in the contracted section. Therefore, the equation in figure 22 was modified to the equation in figure 23 assuming that the two reference elevations were approximately equal (mild slope) and substituting for V2 from the continuity equation.

y sub 2 equals y hat sub 1 plus the quantity V hat sub 1 squared divided by 2g end quantity minus open parenthesis 1 plus C sub cB close parenthesis times the quantity V sub 2 squared divided by 2g end quantity equals y hat sub 1 plus the quantity V hat sub 1 squared divided by 2g end quantity times open bracket 1 minus open parenthesis 1 plus C sub cB close parenthesis times A hat sub 1 squared divided by A sub 2 squared close bracket.

Figure 23. Equation. Modified energy equation for case without riprap.

 

Where:

CcB = Contraction coefficient for model without riprap (dimensionless).

The quantity in brackets is defined in figure 24 as the channel shape factor for the case withoutriprap.

beta sub B equals 0.5 times open bracket 1 minus open parenthesis 1 plus C sub cB close parenthesis times  A hat sub 1 squared divided by A sub 2 squared close bracket.

Figure 24. Equation. Channel shape factor for case without riprap.

 

Where:

βB = Channel shape factor for model without riprap (dimensionless).

Substituting the channel shape factor yielded the equation in figure 25.

y sub 2 divided by y hat sub 1 equals 1 plus beta sub B times Fr hat sub 1 squared.

Figure 25. Equation. Flow depth ratio for case without riprap.

 

Where:

Froud number1 = Froude number for CFD experiments, which is defined as (dimensionless).

An equation for shear stress was also required, as shown in figure 26.

tau sub B equals gamma times y sub 2 times S.

Figure 26. Equation. Average bed shear stress for case without riprap.

 

Where:

S = Energy slope (ft/ft (m/m)).
γ = Unit weight of water (lbf/ft3 (N/m3)).

Using Manning’s equation, as shown in figure 27, and assuming relatively smooth walls so that the hydraulic radius can be approximated as area divided by width (A/W) yielded the equation given in figure 28.

Q equals the quantity 1 divided by n sub B end quantity times A sub 2 times R sub h to the two-thirds power times S to the one half power.

27. Equation. Manning’s equation for case without riprap.

 

Where:

nB = Roughness coefficient for bed material (dimensionless).
Rh = Hydraulic radius (ft (m)).

tau sub B equals gamma times open parenthesis W sub 2 to the one-seventh power divided by A sub 2 close parenthesis to the seven thirds power times open parenthesis Q times n sub B close parenthesis squared.

Figure 28. Equation. Bed shear for case without riprap.

 

Riprap Apron

The same application of the continuity, energy, and momentum equations was applied to the case with riprap. Figure 29 provides a conceptual definition sketch of the approach (upstream) section and contracted section for the case where there was riprap. The flow depth at equilibrium scour, y2R, is shown as greater than the flow depth at equilibrium without the riprap, y2. As for the case without riprap, to simplify the numerical modeling with CFD, an alternative definition sketch was used, as shown in figure 30. The change from the conceptual definition was to alter the upstream section so that the bottom elevation was increased by the estimated contraction scour, yc.

This figure shows a side view of the approach flow with a depth of y sub 1 and illustrates the variables at the abutment, including y sub 0, y sub 2R, and y sub c. A front view shows the abutments on both sides, with the riverbed before scour at a depth of y sub 0 and the contraction scour y sub c. Additional scour below y sub c is shown. The opening width, W sub 2, between the abutments is shown, as well as the width between the riprap aprons, W sub B.
WB = Bottom width of bed material.
Figure 29. Sketch. Conceptual schematic for case with riprap.

 

This figure shows a side view of the approach flow with a depth of y sub 1 plus y sub c and illustrates the variables at the abutment, including y sub 0, y sub 2, and y sub c. A front view shows the abutments on both sides, with the river bed before scour at a depth of y sub 0 and the contraction scour y sub c. The opening width between the riprap aprons, W sub B, is shown.
Figure 30. Sketch. CFD schematic for case with riprap.

 

The flow condition in the opening with riprap at the estimated contraction scour without riprap (y2=y0 + yc) is shown in figure 31. For this case, the bottom roughness for the natural bed material was nB, and the bottom roughness for the riprap was nR.

This figure shows a rectangle with a depth of y sub 2 and a width of W sub 2. In the lower left and lower right of the rectangle are riprap aprons, each with a width of W sub R. The roughness of the riprap is indicated by n sub R, and the roughness of the bed between the riprap aprons is indicated by n sub B. The gap between the aprons is W sub B. The net area, excluding the riprap apron, is A sub 2R.
A2R = Cross-sectional area of contracted section with riprap after contraction scour to a depth of yc.
Figure 31. Schematic. Cross section in the opening for case with riprap.

 

Figure 32 displays a schematic consistent with the CFD analysis for use with the continuity and energy equations. The variable ŷ1 was equal to y1 + yc, as shown in figure 30.

This figure shows an isometric control volume with the approach (section 1) and the contraction (section 2). The variables shown in section 1 are V hat sub 1, A hat sub 1, and y hat sub 1. The variables shown in section 2 are V sub 2R, A sub 2R, and y sub 2.
Figure 32. Schematic. Continuity and energy control volume for case with riprap.

 

For the case with riprap, the continuity equation is summarized in figure 33, and the energy equation is summarized in figure 34.

A hat sub 1 equals V sub 2R times A sub 2R.

Figure 33. Equation. Continuity equation for case with riprap.

 

z sub 1 plus y hat sub 1 plus the quantity alpha sub 1 times V hat sub 1 squared divided by 2g end quantity equals z sub 2 plus y sub 2 plus the quantity alpha sub 2R times V sub 2R squared divided by 2g end quantity plus h sub R.

Figure 34. Equation. Energy equation for case with riprap.

 

Where:

V2R = Average velocity in contracted section with riprap after contraction scour to a depth of yc (ft/s (m/s))
α2R = Energy correction factor for contracted section with riprap (dimensionless).
hR = Head loss between upstream section and contracted section with riprap (ft (m)).

The energy correction factors were approximately equal to 1.00 to 1.01 for gradually varied flow. The head loss between to the two sections, hR, was defined as the contraction coefficient for the model without riprap, CcR, times velocity head in the contracted section. The equation in figure 34 was modified to the equation in figure 35 assuming that the two reference elevations were approximately equal (mild slope) and substituting for V2 from the continuity equation.

y sub 2 equals y hat sub 1 plus the quantity V hat sub 1 squared divided by 2g end quantity minus open parenthesis 1 plus C sub cR close parenthesis times the quantity V sub 2R squared divided by 2g end quantity equals y hat sub 1 plus V hat sub 1 squared divided by 2g times open bracket 1 minus open parenthesis 1 plus C sub cR close parenthesis times A hat sub 1 squared divided by A sub 2R squared close bracket.

Figure 35. Equation. Modified energy equation for case with riprap.

 

To develop a relation for shear stress at the bed after contraction scour had reached a depth of y2, two assumptions were made. First, it was assumed that flow in the contraction could be characterized as gradually varied flow. From this, it followed that the flow acceleration was approximated as 0, and the difference between the upstream pressure and the downstream pressure, PuPd, was also approximately 0. Second, it was assumed that the front faces of abutment were smooth. From this, the shear on the front face of abutment, τa, was approximately 0. These quantities are shown on the control volume schematic in figure 36.

This figure shows an isometric control volume with the approach and the contraction. Several forces applied to the contraction control volume are indicated: upstream pressure, P sub u; downstream pressure, P sub d; the gravitational force caused by g; the average shear, tau sub R; and the riprap shear, tau sub rip. Side shear, tau sub a, is shown equal to 0. The cross-section area, A sub 2R, is shown.
τrip = Average bed shear stress on riprap.
Figure 36. Schematic. Momentum control volume for case with riprap.

 

The average bed shear for the case with riprap in the contraction is shown in figure 37.

tau sub R times open parenthesis W sub 2 minus 2 times W sub R close parenthesis times l sub 2 plus tau sub rip times 2 times W sub R times l sub 2 equals gamma times A sub 2R times l sub 2 times S.

Figure 37. Equation. Bed shear stress for case with riprap.

 

The average bed shear in the opening is expressed in figure 38.

tau sub avg equals open bracket tau sub r times open parenthesis times W sub 2 minus 2 times W sub R close parenthesis plus tau sub rip times 2 times W sub R close bracket divided by W sub 2.

Figure 38. Equation. Average bed shear stress for case with riprap.

 

By substituting the equation in figure 38 into the equation in figure 37, an expression for the energy slope was derived, as shown in the equation in figure 39.

S equals tau sub avg times W sub 2 divided by gamma divided by A sub 2R.

Figure 39. Equation. Energy slope for case with riprap.

 

Manning’s formula in the open channel with composite roughness, n, is written as shown in figure 40.

Q equals 1 divided by n times A sub 2R times R sub h to the two-thirds power times S to the one-half power.

Figure 40. Equation. Manning’s equation for case with riprap.

 

Assuming relatively smooth abutment walls and neglecting the vertical sides of the riprap, the hydraulic radius could be approximated as A2R/W2. Substituting the equation in figure 39 into the equation in figure 40 yields the expression shown in figure 41 for average bottom shear stress.

tau sub avg equals gamma times open parenthesis W sub 2 to the one-seventh power divided by A sub 2R close parenthesis to the seven-thirds power times open parenthesis Q times n close parenthesis squared.

Figure 41. Equation. Bed shear for case with riprap.

 

The composite Manning’s roughness coefficient, n, can be calculated using the composite roughness formula shown in figure 42.(21,22)

n equals open parenthesis 2 times W sub R times n sub R to the 1.5 power plus W sub B times n sub B to the 1.5 power close parenthesis divided by W sub 2 close parenthesis to the two-thirds power. 

Figure 42. Equation. Composite n.

 

Composite n for the bed and riprap materials was calculated using Strickler’s equation, which relates roughness coefficient to the size of bed material.(23) Following the HEC-23 design guidance, the apron width, WR, was specified as 2y0. Expressing the natural bed width in terms of W2 led to the representation of composite n in figure 43.(3)

n equals open parenthesis 4 times y sub 0 times n sub R to the 1.5 power plus open parenthesis W sub 2 minus 4 times  y sub 0 close parenthesis times n sub B to the 1.5 power close parenthesis divided by  W sub 2 close parenthesis to the two-thirds power.
Figure 43. Equation. Composite n in terms of depth and width.

 

Equivalent Depth for the Riprap Apron

 

The section in figure 31 for the riprap case was simplified to the equivalent rectangular cross section shown in figure 44. The equivalent depth, ye, plus the vertical adjustment, Δ Z, was equal to y2. The bed had a composite roughness, n, and an average bed shear stress of τavg. The cross-sectional area was the same as it was for the previous riprap case.

This figure shows a shaded rectangle with depth of y sub e and width of W sub 2. The area is indicated as A sub 2R. The roughness, n, and the shear tau sub avg are shown. Below the rectangle at the same width is a smaller rectangle with depth equal to delta Z.
Figure 44. Schematic. Equivalent opening cross section with uniform roughness and shear.

 

Figure 45 illustrates the control volume for the equivalent depth case with riprap. The energy equation for this case is shown in figure 46.

This figure shows an isometric control volume with the approach (section 1) and the contraction (section 2). The variables shown in section 1 are V hat sub 1, A hat sub 1, and y hat sub 1. The variables shown in section 2 are V sub 2R, A sub 2R, y sub 2, delta Z, and the average roughness, n.
Figure 45. Schematic. Control volume for the equivalent depth with riprap.

 

z sub 1 plus y hat sub 1 plus the quantity alpha sub 1 times V hat sub 1 squared divided by 2g end quantity equals z sub 2 plus y sub 2 plus the quantity alpha sub 2R times V sub 2R squared divided by 2g end quantity plus h sub R equals z sub 2 plus y sub e plus the quantity alpha sub 2e times V sub 2R squared divided by 2g end quantity plus h sub e plus delta Z.

Figure 46. Equation. Energy equation for equivalent depth case with riprap.

 

Assuming a mild slope where z1 was approximately equal to z2 and taking the energy correction factors α2R and α2e as equal to 1.0 led to the equation shown in figure 47.

y sub e equals y hat sub 1 plus the quantity V hat sub 1 squared divided by 2g end quantity minus open parenthesis 1 plus C sub ce close parenthesis times the quantity V sub 2R squared divided by 2g end quantity minus delta Z equals y hat sub 1 plus the quantity V hat sub 1 squared divided by 2g end quantity times open bracket 1 minus open parenthesis 1 plus C sub ce close parenthesis times the quantity A hat sub 1 squared divided by A sub 2R squared end quantity close bracket minus delta Z.

Figure 47. Equation. Modified energy equation for equivalent depth case with riprap.

 

Where:

Cce = Contraction coefficient for equivalent model with riprap (dimensionless).

Defining the channel shape factor, βe, as shown in figure 48, and rewriting the equation in figure 47 in terms of a depth ratio led to the equation in figure 49.

Beta sub e equals 0.5 times open bracket 1 minus open parenthesis 1 plus C sub ce close parenthesis times A hat sub 1 squared divided by A sub 2R squared close bracket.

Figure 48. Equation. Channel shape factor for equivalent depth case with riprap.

 

y sub e divided by y hat sub 1 equals 1 plus beta sub e times Fr hat sub 1 squared minus the quantity delta Z divided by y hat sub 1.

Figure 49. Equation. Depth ratio for equivalent depth case with riprap.

 

Where:

Βe = Channel shape factor for equivalent model with riprap (dimensionless).

Shear Stress Ratio

The ratio of bed shear for the case with riprap (figure 41) to the one without riprap (figure 28) is shown in figure 50.

tau sub avg divided by tau sub B equals open parenthesis A sub 2 divided by A sub 2R close parenthesis to the seven-thirds power open parenthesis n divided by n sub b close parenthesis squared.

Figure 50. Equation. Shear stress ratio.

 

Using the expression for composite n in figure 43, the ratio of roughness values is shown in the equation in figure 51.

n divided by n sub B equals open bracket the quantity 4 times y sub 0 times n sub R to the 1.5 power plus open parenthesis W sub 2 minus 4 times y sub 0 close parenthesis times n sub B to the 1.5 power end quantity divided by W sub 2 close bracket to the two-thirds power times 1 divided by n sub B equals open bracket the quantity 4 times n sub R to the 1.5 power plus open parenthesis W sub 2 divided by y sub 0 minus 4 close parenthesis times n sub B to the 1.5 power end quantity divided by the quantity W sub 2 divided by y sub 0 end quantity close bracket to the two-thirds power times 1 divided by n sub B.

Figure 51. Equation. Manning’s roughness ratio.

 

The area ratio in figure 50 can be expressed in terms of depth. Because A2 was equal to W2 times y2 (figure 19), and A2R was equal to W2 times ye (figure 44), the area ratio is represented as shown in figure 52.

A sub 2 divided by A sub 2R equals y sub 2 divided by y sub e.

Figure 52. Equation. Area ratio.

 

Substituting the roughness ratio in figure 51 and the area ratio in figure 52 into the equation in figure 50 led to the shear stress ratio in figure 53.

tau sub avg divided by tau sub B equals open parenthesis y sub 2 divided by y sub e close parenthesis to the seven-thirds power open parenthesis open bracket the quantity 4 times n sub R exponent 1.5 plus open parenthesis W sub 2 divided by y sub 0 minus 4 close parenthesis times n sub B to the 1.5 power end quantity divided by the quantity W sub 2 divided by y sub 0 end quantity close bracket to the two-thirds power times 1 divided by n sub B close parenthesis squared.

Figure 53. Equation. Shear stress ratio with substitutions.

 

Further substitution for y2 (figure 25) and ye (figure 49) led to the equation in figure 54.

tau sub avg divided by tau sub B equals open bracket the quantity 4 times open parenthesis n sub R divided by n sub B close parenthesis to the 1.5 power plus open parenthesis the quantity W sub 2 divided by y sub 0 end quantity minus 4 close parenthesis end quantity divided by the quantity W sub 2 divided by y sub 0 end quantity close bracket to the four-thirds power open parenthesis the quantity 1 plus beta sub B times FR hat sub 1 squared end quantity divided by the quantity 1 plus beta sub e times FR hat sub 1 squared minus the quantity delta Z divided by y hat sub 1 end quantity end quantity close parenthesis to the seven-thirds power.

Figure 54. Equation. Shear stress ratio with average shear stress.

 

Where:

Froud number = Froude number for CFD experiments, which is defined as (dimensionless).

On the left side of the equation, τavg was replaced with τR, and on the right side, the exponent of  4/3 was replaced by a new parameter, ε, as shown in figure 55. A relation for estimating ε was explored by evaluating the CFD analytical results.

tau sub R divided by tau sub B equals open bracket the quantity 4 times open parenthesis n sub R divided by n sub B close parenthesis to the 1.5 power plus open parenthesis the quantity W sub 2 divided by y sub 0 end quantity minus 4 close parenthesis end quantity divided by the quantity W sub 2 divided by y sub 0 end quantity close bracket to epsilon power times open parenthesis the quantity 1 plus beta sub B times FR hat sub 1 squared end quantity divided by the quantity 1 plus beta sub e times FR hat sub 1 squared minus the quantity delta Z divided by y hat sub 1 end quantity end quantity to the seven-thirds power.

Figure 55. Equation. Shear stress ratio with τR

 

Where:

ε = Function related to ratio of roughness of riprap to that of erodible bed material (dimensionless).

As is apparent from the equation in figure 55, the ratio of the shear stress in the middle of the channel with and without the riprap aprons varied with several factors. The relative size of the riprap relative to the bed material was expressed in the ratio of the Manning’s roughness values, nR/nB. The flow dimensions in the contraction were expressed in the ratio of the contraction width to the flow depth before scour. A high value of this ratio could be considered a wide channel, and a low value might be called a narrow channel. In addition, the degree of contraction, including energy losses, was included in the term in the equation that included the channel shape factors. The extent of contraction would also influence the size of riprap required for abutment protection. This shear ratio was investigated with the CFD modeling described in chapter 5.

 

 

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