https://orcid.org/0000-0001-7060-8378
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Abstract
A numerically-based approach is used to obtain a size selection formula for the riprap stone part of the apron used to protect wing-wall abutments against erosion. Several series of simulations with a given mean riprap diameter are conducted to estimate the maximum bed shear stress over the riprap apron and then the maximum (critical) Froude number at which riprap stones will resist shear failure by the flow for cases where the abutment is placed in straight and curved channels with or without a floodplain. Simulation results show that some of the existing design formulas for wing-wall abutments placed in straight river channels are not conservative enough for relatively large floodplain widths. Using data from the numerical simulations, a modified, two parameter design formula for riprap size selection is proposed that explicitly incorporates the effects of bank curvature and floodplain width.
Acknowledgement
This research was funded by Grant # 15801350/15243500/69A3551747107 from the Mid-America Transportation Center (MATC). The authors would like to thank MATC for supporting this research.
Notation
| Bf | = | width of the floodplain (m) |
| C | = | model coefficient (–) |
| D50 | = | mean riprap stone diameter (m) |
| d50 | = | mean sand diameter (m) |
| Fr | = | Froude number (–) |
| Href | = | length scale (m) |
| g | = | gravitational acceleration (m s−2) |
| Ks | = | shape factor (–) |
| ks | = | surface roughness (m) |
| P | = | pressure (Pa) |
| R | = | channel radius of curvature (m) |
| Re | = | Reynolds number (–) |
| R4 | = | width of the riprap apron (m) |
| Ss | = | specific gravity of the riprap stone (–) |
| Ui | = | velocity component along the i direction (m s−1) |
| uτc | = | critical friction velocity for riprap shear failure (m s−1) |
| V | = | velocity scale (m s−1) |
| W | = | width of main channel (m) |
| xi | = | space coordinate the i direction (m) |
| yf | = | flow depth over the floodplain (m) |
| ym | = | flow depth over the main channel (m) |
| α | = | power law exponent (–) |
| δik | = | Kronecker symbol (–) |
| τ | = | bed shear stress (Pa) |
| ρ | = | fluid density (kg m−3) |
| μ | = | dynamic viscosity (kg m−1 s−1) |
| μt | = | eddy viscosity (kg m−1 s−1) |
Correction Statement
This article has been corrected with minor changes. These changes do not impact the academic content of the article.