IV. RESULTADOS
4.1. Diagnóstico Situacional
4.1.2. Cantidades de RAEE
The other main input parameter in the HEC-RAS geometric models were the roughness coefficient to be used on the models. There were two options that the HEC-RAS model allowed the user to choose from. The user could either choose to use the Manning n roughness coefficient or the ks roughness coefficient.
The Manning n roughness coefficient can be determined from design manuals or literature. Using literature or design manuals as a reference for choosing Manning n roughness coefficient requires that the designer chooses a value of the Manning n for specific bed material types. The main reasoning behind the procedure is that the materials of the same type and physical roughness should comprise the same roughness, thus the same Manning n roughness coefficient. Then a calibration may be done to ensure that more accurate results are computed with the HEC-RAS numerical model.
Alternatively, the roughness element height ks could be chosen to define the roughness of the bed. The roughness coefficient ks is generally unique for each rough bed and is more
6-141 representative of the roughness conditions of each unique bed condition. Ks can be understood as the size of the eddies that form due to the roughness of the bed (Langmaak, 2013). It was impossible to determine one unified mathematical definition for the ks value. Froehlich (2012) also identified the problem in the definition of the ks value, then defined ks in the following manner.
𝑘𝑠 = 𝛼𝑖𝐷𝑖 Eq. 61
Whereby 𝐷𝑖 is the diameter of the particle that is larger than the percentage i by mass (it could be D50 or D90). The 𝛼𝑖 is the constant associated with the 𝐷𝑖 Froehlich (2012). Froelich agrees that there are different constants and 𝐷𝑖 defined in different literature in which Bray (1982),
Maynord (1991)) summarise some of the analyses which assess the value of ks. Froelich(2012) noted that the main values that are generally used for the 𝐷𝑖 to define ks are D50, D65, D84 and D90.
In a similar study by Langmaak (2015) the value of the constant 𝛼𝑖 was determined for two gradings and the value of 0.81 was obtained from a calibration. The D90 was used, as a result, Equation 61 was applicable in the study of large riprap in rough turbulent flow conditions:
𝑘𝑠 = 0.81 𝐷90 Eq. 62 Alternatively, the well-known Chezy (1769) equation could be used to calculate the
roughness height, 𝑘𝑠. The Chezy equation was defined as follows (Huthoff and Augustijn,
2004)): 𝐶 = 18 log (12𝑅 𝐾𝑠 ) Eq. 63 Whereby, 𝐶 = 𝑉 √𝑅𝑆𝑜 Eq. 64 R= Hydraulic Radius
C= Chezy roughness coefficient
V= channel average flow velocity
6-142 The Chezy equation was chosen as a reliable mathematical expression to use to determine the relevant ks value for this study. However, a reliable laboratory test was needed as a reference for the calibration of the HEC-RAS model.
The most reliable test was considered as the test that showed the minimum MN standard deviation to the mean. The test with the minimum standard deviation of MN was determined from Appendix F. Hence, test P2M2T3 was used to calibrate the HEC-RAS model in order to determine the relevant ks value. Test P2M2T3 had a standard deviation of 0.013 for the MN, relative to the mean.
To determine the ks value to be used, Table 21 was used for the calibration. The flow conditions and boundary conditions for the test P2M2T3 were modelled, as well as the geometry with the 0.4 steep bed slope. An initial guess of the ks roughness value was applied on the steep downslope of the main testing area. The initial guess was chosen to be ks = 0.5*0.135= 0.0675, this was based on Equation 62 and the initial assumption made was that, 𝛼90 = 0.5 and 𝐷90 = 0.135 m.
Thereafter, the HEC-RAS simulation was run and the wetted perimeter, flow area, average velocity and Froude numbers were simulated by the HEC-RAS model based on the initially assumed ks value. For each cross-section in the downslope testing area, the hydraulic radius was calculated as the ratio of the flow area to the wetted perimeter. Then the Chezy coefficient was calculated based on the simulated results using the initially guessed ks value. Then on the second last column in Table 21, the Chezy coefficient was calculated using solver (solver computed the ks values that will produce the C value calculated based on the initially assumed ks). Then the ks value on the last column changed to a different value than the initially assumed. For the next iteration, the same method was followed and the ks value converged down until the ks values on the last column corresponded with the ks values on the HEC-RAS model. The final calibrated results are summarised in Table 21.
6-143 Table 21: Final Chezy equation ks determination value.
Cross- section no. Wetted perimeter (m) Flow area (m2) Hydraulic radius Velocity (m/s) Froude number C (calculated from velocity in channel) C (based on ks)
ks
1 0.74 0.02 0.0270 1.86 3.13 17.889 17.889 0.0329 2 0.74 0.02 0.0270 1.9 3.23 18.274 18.274 0.0313 3 0.73 0.02 0.0274 1.93 3.31 18.436 18.436 0.0311 4 0.73 0.02 0.0274 1.96 3.39 18.723 18.723 0.0300 5 0.73 0.02 0.0274 1.99 3.47 19.009 19.009 0.0289 6 0.73 0.02 0.0274 2.02 3.54 19.296 19.296 0.0279 7 0.73 0.02 0.0274 2.05 3.61 19.583 19.583 0.0269 8 0.73 0.02 0.0274 2.08 3.68 19.869 19.869 0.0259 9 0.73 0.02 0.0274 2.1 3.74 20.060 20.060 0.0253 10 0.72 0.02 0.0278 2.12 3.8 20.112 20.112 0.0254 11 0.72 0.02 0.0278 2.14 3.85 20.302 20.302 0.0248 12 0.72 0.02 0.0278 2.17 3.93 20.586 20.586 0.0239 13 0.72 0.02 0.0278 2.18 3.96 20.681 20.681 0.0237The final calibrated ks values in Table 21, on the last column, were not all equal. The ks calculation was based on the cross-section position. The cross-section at the top (closer to the crest of the hydraulic model) had higher ks values due to lower velocities at the top. The high ks values were located at the bottom cross-sections where the velocities were high. Therefore, the ks value at the bottom was chosen as the critical ks and chosen as the relevant ks value to be used for the rest of the HEC-RAS simulations. Since it was assumed that the grading and roughness of the riprap were the same for all the tests of the same median stone diameter and grading. Consequently, it was assumed that the following relationship holds 𝛼50 =
𝑘𝑠
𝐷50 =
0.0237
0.075 = 0.316.
So, the same ks values were used for Test series two and Test series three HEC-RAS model tests since the D50 median stone size was the same. The ks values used to simulate the Test series two and Test series three tests were 0.0237 in the downslope testing area. However, for the Test series one HEC-RAS simulations, the ks value applied on the downslope test area was 0.012, since 𝛼50 = 0.316.
Figure 62 shows a typical cross-section taken from the downslope area and the roughness ks value applied to the bed area.
6-144 Figure 62: Typical cross-section extracted along the half trapezoidal steep downslope.
To define the ks values for the wall and the bed area, the bank station range (0.001-1.99) were used to indicate where the ks should be applied. The top black fine line shows the value (circled in red) and ks value applied.
With all the geometric data, boundary conditions, flow rates at incipient failure and roughness element values available, it was possible to simulate and obtain reliable results from the 1-D HEC-RAS simulations.