Stone Stability Under Non-Uniform Flow- Sự ổn định của viên đá gia cố đáy dưới tác động của dòng chảy không đều

17.9 10.0 0.43 0.054 0.093 20.49 2.56 14.69 8.115E-07 1KR 32.0 18.0 9.1 0.38 0.049 0.077 16.05 2.01 11.66 2.396E-07 1LR 35.5 19.0 10.0 0.39 0.052 0.087 19.27 2.47 13.74 5.446E-07 Profile 2 (∆ = 0.341) 1AR 22.0 12.1 5.5 0.42 0.052 0.100 18.94 2.34 12.90 1.485E-06 1BR 20.0 12.1 5.0 0.39 0.043 0.069 15.60 1.93 10.75 3.005E-07 1CR 23.0 13.0 5.7 0.40 0.051 0.094 18.02 2.24 12.16 6.300E-07 1DR 26.5 14.3 6.6 0.40 0.052 0.100 19.19 2.35 13.40 1.208E-06 1ER 24.0 13.9 6.0 0.37 0.047 0.081 16.51 2.06 11.20 3.468E-07 1FR 27.0 14.8 6.7 0.38 0.047 0.082 17.67 2.21 12.15 4.566E-07 1GR 31.0 16.1 7.7 0.39 0.056 0.113 20.38 2.58 14.16 1.179E-06 1HR 28.0 15.9 7.0 0.36 0.048 0.082 16.68 2.11 11.51 5.028E-07 1IR 31.5 17.1 7.9 0.36 0.051 0.096 17.95 2.27 12.85 6.011E-07 1JR 35.5 18.1 8.9 0.37 0.055 0.109 19.95 2.57 14.01 1.266E-06 1KR 32.0 18.1 8.0 0.33 0.050 0.091 16.01 2.03 11.80 2.601E-07 1LR 35.5 19.1 8.9 0.34 0.051 0.095 18.79 2.44 13.29 6.820E-07 Profile 3 (∆ = 0.341) 1AR 22.0 12.1 5.2 0.39 0.050 0.090 18.01 2.16 12.01 1.272E-06 1BR 20.0 12.2 4.7 0.36 0.041 0.063 14.36 1.73 9.75 2.023E-07 1CR 23.0 13.2 5.4 0.36 0.046 0.078 16.10 1.93 10.93 2.774E-07 1DR 26.5 14.5 6.2 0.36 0.050 0.091 17.97 2.22 12.44 4.913E-07 1ER 24.0 14.1 5.6 0.34 0.046 0.077 14.89 1.79 10.23 3.352E-07 1FR 27.0 15.2 6.3 0.35 0.049 0.089 16.39 2.04 11.23 2.601E-07 1GR 31.0 16.2 7.3 0.36 0.053 0.104 18.83 2.35 13.42 1.087E-06 1HR 28.0 16.2 6.6 0.32 0.047 0.080 15.20 1.91 10.55 2.081E-07 1IR 31.5 17.4 7.4 0.33 0.051 0.094 16.10 2.03 11.62 4.393E-07 1JR 35.5 18.3 8.3 0.34 0.055 0.111 18.13 2.27 12.67 4.277E-07 1KR 32.0 18.3 7.5 0.31 0.052 0.101 14.62 1.83 10.74 2.196E-07 1LR 35.5 19.1 8.3 0.32 0.051 0.096 17.03 2.19 11.99 3.179E-07 Profile 4 (∆ = 0.341) 1AR 22.0 12.3 4.9 0.36 0.050 0.091 16.12 1.95 11.03 4.913E-07 1BR 20.0 12.3 4.4 0.33 0.039 0.056 12.80 1.55 8.76 1.561E-07 1CR 23.0 13.3 5.1 0.34 0.046 0.076 14.44 1.79 9.95 2.485E-07 1DR 26.5 14.5 5.9 0.34 0.050 0.089 15.88 1.96 11.26 4.277E-07 1ER 24.0 14.3 5.3 0.32 0.044 0.069 13.55 1.68 9.35 1.329E-07 1FR 27.0 15.3 6.0 0.32 0.045 0.075 14.69 1.83 10.22 1.676E-07 1GR 31.0 16.2 6.9 0.34 0.050 0.090 16.70 2.07 11.59 5.202E-07 1HR 28.0 16.5 6.2 0.30 0.045 0.073 13.28 1.67 9.71 8.091E-08 1IR 31.5 17.8 7.0 0.30 0.045 0.072 14.58 1.84 10.52 1.849E-07 1JR 35.5 18.5 7.8 0.32 0.049 0.088 16.19 2.03 11.39 4.797E-07 1KR 32.0 18.5 7.1 0.29 0.048 0.084 13.29 1.69 9.75 1.156E-07 1LR 35.5 19.3 7.8 0.30 0.047 0.082 15.02 1.92 10.72 2.312E-07 126 Appendix B. Data Table B.2: Summary of measured governing variables (set-up 2, α = 50). Q h Re Fr u∗ ΨS ΨWL ΨLm Ψu−σ ΦE [l/s] [cm] [104] [-] [m/s] [-] [-] [-] [-] [-] Profile 1 (∆ = 0.320) 2AR 22.0 11.6 6.2 0.51 0.046 0.082 20.55 2.56 14.03 1.790E-06 2BR 20.0 12.0 5.7 0.44 0.043 0.072 16.49 2.03 11.22 6.623E-07 2CR 23.0 12.8 6.5 0.46 0.048 0.088 21.57 2.64 14.86 1.366E-06 2DR 26.5 13.8 7.5 0.47 0.051 0.102 23.55 2.96 17.12 1.623E-06 2ER 24.0 13.2 6.8 0.46 0.046 0.083 19.93 2.47 13.88 9.904E-07 2FR 27.0 14.4 7.6 0.45 0.048 0.090 21.35 2.67 15.29 1.092E-06 2GR 31.0 16.0 8.8 0.44 0.051 0.102 22.26 2.90 16.08 1.593E-06 2HR 28.0 15.9 7.9 0.40 0.049 0.092 19.01 2.39 13.63 7.219E-07 2IR 31.5 16.9 8.9 0.41 0.050 0.097 20.56 2.58 14.88 9.188E-07 2JR 35.5 17.5 10.0 0.44 0.051 0.100 22.73 2.89 16.78 1.104E-06 2KR 32.0 17.8 9.1 0.39 0.047 0.085 18.27 2.32 13.60 1.354E-06 2LR 35.5 18.6 10.0 0.41 0.051 0.100 21.48 2.75 15.48 9.725E-07 Profile 1a (∆ = 0.384) 2AR 22.0 11.6 6.2 0.51 0.046 0.068 17.12 2.13 11.69 6.536E-07 2BR 20.0 12.0 5.7 0.44 0.043 0.060 13.75 1.70 9.35 1.743E-07 2CR 23.0 12.8 6.5 0.46 0.048 0.074 17.97 2.20 12.38 5.555E-07 2DR 26.5 13.8 7.5 0.47 0.051 0.085 19.63 2.46 14.27 6.754E-07 2ER 24.0 13.2 6.8 0.46 0.046 0.069 16.61 2.05 11.57 3.268E-07 2FR 27.0 14.4 7.6 0.45 0.048 0.075 17.79 2.22 12.74 5.174E-07 2GR 31.0 16.0 8.8 0.44 0.051 0.085 18.55 2.41 13.40 7.026E-07 2HR 28.0 15.9 7.9 0.40 0.049 0.077 15.84 1.99 11.36 4.030E-07 2IR 31.5 16.9 8.9 0.41 0.050 0.080 17.13 2.15 12.40 4.575E-07 2JR 35.5 17.5 10.0 0.44 0.051 0.083 18.94 2.41 13.98 4.629E-07 2KR 32.0 17.8 9.1 0.39 0.047 0.071 15.22 1.93 11.33 2.832E-07 2LR 35.5 18.6 10.0 0.41 0.051 0.084 17.90 2.29 12.90 5.446E-07 Profile 2 (∆ = 0.341) 2AR 22.0 11.6 5.6 0.46 0.049 0.088 18.07 2.16 12.19 6.473E-07 2BR 20.0 11.9 5.1 0.40 0.046 0.077 14.21 1.74 9.62 2.196E-07 2CR 23.0 12.6 5.8 0.42 0.050 0.091 18.30 2.23 12.42 5.780E-07 2DR 26.5 13.8 6.7 0.42 0.057 0.119 20.96 2.59 14.60 8.091E-07 2ER 24.0 13.2 6.1 0.41 0.050 0.091 17.33 2.15 11.75 5.780E-07 2FR 27.0 14.1 6.9 0.42 0.051 0.096 18.61 2.32 12.89 4.624E-07 2GR 31.0 15.9 7.9 0.40 0.056 0.112 19.23 2.40 13.84 1.133E-06 2HR 28.0 15.8 7.1 0.37 0.051 0.094 16.76 2.08 11.83 4.855E-07 2IR 31.5 16.7 8.0 0.38 0.053 0.104 18.02 2.29 13.00 5.144E-07 2JR 35.5 18.0 9.0 0.38 0.055 0.111 19.99 2.54 14.60 6.936E-07 2KR 32.0 17.9 8.1 0.35 0.049 0.089 16.09 2.03 11.86 3.121E-07 2LR 35.5 18.2 9.0 0.38 0.053 0.101 18.59 2.36 13.32 5.028E-07 Profile 3 (∆ = 0.341) 2AR 22.0 11.6 5.1 0.42 0.048 0.085 16.10 1.89 10.92 4.393E-07 2BR 20.0 11.8 4.7 0.37 0.039 0.056 12.86 1.53 8.75 1.272E-07 2CR 23.0 12.7 5.4 0.38 0.049 0.088 16.22 1.96 11.15 2.601E-07 2DR 26.5 13.7 6.2 0.39 0.053 0.104 18.51 2.25 12.95 5.491E-07 2ER 24.0 13.3 5.6 0.37 0.049 0.087 15.56 1.89 10.66 3.237E-07 2FR 27.0 14.2 6.3 0.38 0.052 0.097 17.03 2.11 11.76 3.294E-07 2GR 31.0 15.9 7.2 0.37 0.054 0.106 17.13 2.11 12.41 5.780E-07 2HR 28.0 15.9 6.5 0.33 0.049 0.089 15.24 1.88 10.84 3.005E-07 2IR 31.5 16.6 7.3 0.35 0.052 0.097 16.22 2.02 11.73 3.121E-07 2JR 35.5 17.9 8.3 0.35 0.056 0.116 18.08 2.27 13.34 4.797E-07 2KR 32.0 17.8 7.5 0.32 0.050 0.091 14.69 1.88 11.04 1.676E-07 2LR 35.5 18.5 8.3 0.34 0.054 0.108 16.97 2.18 12.18 3.872E-07 Profile 4 (∆ = 0.341) 2AR 22.0 11.8 4.7 0.38 0.046 0.079 14.05 1.67 9.70 2.370E-07 2BR 20.0 11.8 4.3 0.34 0.041 0.060 11.30 1.33 7.75 8.669E-08 2CR 23.0 12.9 5.0 0.35 0.047 0.081 14.26 1.72 9.89 2.023E-07 2DR 26.5 13.9 5.7 0.36 0.051 0.096 16.24 1.96 11.49 4.335E-07 2ER 24.0 13.3 5.2 0.34 0.047 0.082 13.69 1.66 9.49 2.081E-07 2FR 27.0 14.3 5.8 0.35 0.049 0.087 14.66 1.80 10.40 2.138E-07 2GR 31.0 16.1 6.7 0.34 0.051 0.096 14.93 1.85 11.00 3.641E-07 2HR 28.0 15.9 6.0 0.31 0.048 0.084 13.25 1.64 9.52 1.618E-07 2IR 31.5 16.8 6.8 0.32 0.049 0.088 14.08 1.75 10.52 1.965E-07 2JR 35.5 18.1 7.6 0.32 0.053 0.102 15.94 2.04 12.03 2.774E-07 2KR 32.0 17.9 6.9 0.29 0.046 0.078 12.73 1.62 9.71 1.734E-07 2LR 35.5 18.6 7.6 0.31 0.051 0.094 14.71 1.91 10.88 2.890E-07 B.3. Governing variables 127 Table B.3: Summary of measured governing variables (set-up 3, α = 70). Q h Re Fr u∗ ΨS ΨWL ΨLm Ψu−σ ΦE [l/s] [cm] [104] [-] [m/s] [-] [-] [-] [-] [-] Profile 1 (∆ = 0.341) 3AR 22.0 12.1 6.2 0.47 0.048 0.085 19.10 2.31 13.02 6.647E-07 3BR 20.0 12.0 5.7 0.44 0.042 0.065 15.93 1.97 10.94 3.872E-07 3CR 23.0 12.9 6.5 0.45 0.049 0.089 20.16 2.47 13.99 6.589E-07 3DR 26.5 13.8 7.5 0.47 0.050 0.092 21.06 2.56 15.06 1.341E-06 3ER 24.0 14.1 6.8 0.41 0.044 0.071 16.92 2.11 11.90 2.948E-07 3FR 27.0 14.9 7.6 0.43 0.048 0.084 19.10 2.38 13.73 5.259E-07 3GR 31.0 15.7 8.8 0.46 0.049 0.089 20.87 2.60 15.32 1.607E-06 3HR 28.0 15.8 7.9 0.41 0.046 0.079 17.82 2.23 13.44 8.034E-07 3IR 31.5 16.9 8.9 0.41 0.049 0.086 19.76 2.48 14.62 1.144E-06 3JR 35.5 17.5 10.0 0.44 0.050 0.091 23.03 2.90 16.50 2.283E-06 3KR 32.0 17.7 9.1 0.39 0.047 0.080 19.16 2.44 14.28 9.132E-07 3LR 35.5 18.3 10.0 0.41 0.051 0.095 21.30 2.74 14.97 1.485E-06 Profile 1a (∆ = 0.384) 3AR 22.0 12.1 6.2 0.47 0.048 0.075 16.96 2.05 11.56 8.061E-07 3BR 20.0 12.0 5.7 0.44 0.042 0.057 14.15 1.75 9.72 4.793E-07 3CR 23.0 12.9 6.5 0.45 0.049 0.079 17.90 2.19 12.42 7.897E-07 3DR 26.5 13.8 7.5 0.47 0.050 0.081 18.70 2.27 13.38 9.804E-07 3ER 24.0 14.1 6.8 0.41 0.044 0.063 15.03 1.88 10.57 2.723E-07 3FR 27.0 14.9 7.6 0.43 0.048 0.075 16.96 2.11 12.19 8.714E-07 3GR 31.0 15.7 8.8 0.46 0.049 0.079 18.54 2.31 13.60 1.095E-06 3HR 28.0 15.8 7.9 0.41 0.046 0.070 15.83 1.98 11.93 5.501E-07 3IR 31.5 16.9 8.9 0.41 0.049 0.076 17.55 2.20 12.99 8.496E-07 3JR 35.5 17.5 10.0 0.44 0.050 0.081 20.45 2.58 14.66 2.353E-06 3KR 32.0 17.7 9.1 0.39 0.047 0.071 17.01 2.16 12.68 6.971E-07 3LR 35.5 18.3 10.0 0.41 0.051 0.084 18.91 2.44 13.29 7.952E-07 Profile 1b (∆ = 0.320) 3AR 22.0 12.1 6.2 0.47 0.048 0.090 20.36 2.46 13.87 1.223E-06 3BR 20.0 12.0 5.7 0.44 0.042 0.069 16.98 2.10 11.66 6.682E-07 3CR 23.0 12.9 6.5 0.45 0.049 0.095 21.48 2.63 14.91 1.283E-06 3DR 26.5 13.8 7.5 0.47 0.050 0.098 22.45 2.73 16.05 1.605E-06 3ER 24.0 14.1 6.8 0.41 0.044 0.075 18.03 2.25 12.68 5.370E-07 3FR 27.0 14.9 7.6 0.43 0.048 0.090 20.35 2.54 14.63 1.617E-06 3GR 31.0 15.7 8.8 0.46 0.049 0.095 22.24 2.77 16.32 2.160E-06 3HR 28.0 15.8 7.9 0.41 0.046 0.084 18.99 2.38 14.32 1.432E-06 3IR 31.5 16.9 8.9 0.41 0.049 0.092 21.05 2.64 15.58 2.058E-06 3JR 35.5 17.5 10.0 0.44 0.050 0.097 24.54 3.10 17.59 3.472E-06 3KR 32.0 17.7 9.1 0.39 0.047 0.085 20.41 2.60 15.21 1.784E-06 3LR 35.5 18.3 10.0 0.41 0.051 0.101 22.70 2.93 15.95 2.727E-06 Profile 2 (∆ = 0.341) 3AR 22.0 12.6 5.4 0.39 0.050 0.091 16.79 2.01 11.56 2.427E-07 3BR 20.0 12.7 5.0 0.35 0.045 0.074 14.17 1.70 9.70 2.138E-07 3CR 23.0 13.3 5.7 0.38 0.049 0.089 16.57 2.07 11.70 4.393E-07 3DR 26.5 14.4 6.6 0.39 0.053 0.103 18.94 2.32 13.11 7.860E-07 3ER 24.0 14.7 5.9 0.34 0.046 0.076 14.86 1.85 10.25 1.561E-07 3FR 27.0 15.5 6.7 0.36 0.051 0.094 17.00 2.09 11.95 4.970E-07 3GR 31.0 16.4 7.7 0.37 0.054 0.106 18.56 2.33 13.30 6.820E-07 3HR 28.0 16.4 6.9 0.34 0.052 0.097 16.09 2.02 11.58 5.086E-07 3IR 31.5 17.8 7.8 0.34 0.055 0.109 17.64 2.21 13.04 6.011E-07 3JR 35.5 18.0 8.8 0.37 0.056 0.116 20.77 2.65 14.94 1.578E-06 3KR 32.0 18.0 7.9 0.34 0.052 0.098 16.97 2.14 12.42 6.358E-07 3LR 35.5 19.2 8.8 0.34 0.053 0.101 18.33 2.35 13.41 7.051E-07 Profile 3 (∆ = 0.341) 3AR 22.0 12.8 4.8 0.34 0.049 0.086 14.43 1.74 10.14 2.832E-07 3BR 20.0 13.0 4.4 0.30 0.042 0.064 11.84 1.43 8.39 9.247E-08 3CR 23.0 13.4 5.1 0.33 0.050 0.092 14.61 1.83 10.19 3.121E-07 3DR 26.5 14.7 5.8 0.33 0.053 0.103 16.29 1.99 11.54 3.757E-07 3ER 24.0 15.0 5.3 0.29 0.045 0.072 12.79 1.62 8.99 1.387E-07 3FR 27.0 15.6 5.9 0.31 0.051 0.097 14.73 1.82 10.50 2.774E-07 3GR 31.0 16.8 6.8 0.32 0.056 0.113 16.04 2.01 11.74 7.918E-07 3HR 28.0 16.6 6.2 0.29 0.052 0.100 13.79 1.77 10.12 2.890E-07 3IR 31.5 17.5 6.9 0.31 0.054 0.108 15.36 1.95 11.54 5.202E-07 3JR 35.5 18.5 7.8 0.32 0.056 0.115 18.00 2.34 13.15 8.901E-07 3KR 32.0 18.2 7.0 0.29 0.054 0.107 14.76 1.88 10.98 2.312E-07 3LR 35.5 19.5 7.8 0.29 0.056 0.116 15.96 2.09 11.89 3.699E-07 128 Appendix B. Data Appendix C Numerical flow modeling C.1 Turbulence modeling C.1.1 Mean-flow equations The flow of an incompressible, viscous Newtonian fluid can be described by a system of flow equations consisting of a continuity equation ∂ui ∂xi = 0 (C.1) and three momentum equations, the so-called Navier Stokes equations ∂ui ∂t + uj ∂ui ∂xj = −1 ρ ∂p ∂xi + ν ∂2ui ∂x2j + fi (C.2) where t is time, xi are spatial coordinates, ui are components of the velocity vec- tor, fi are components of an external force per unit mass, p is the pressure, ρ is the fluid density and ν is the kinematic viscosity. Additional information of this set of flow equations can be found in, for example, Hinze (1975); Rodi (1993). It is in principal possible to solve this set of equations if we know the bound- ary conditions and the initial conditions. However, solving these equations for general turbulent flows requires a very fine computational time- and space-grid to resolve all the scales present in the turbulence motion. These requirements are still far beyond the capacity of the modern computer in term of storage and computational time. Engineers are usually not interested in the details of the fluctuating motion, but in the mean flow field. Therefore a statistical approach can be used in which the Navier Stokes equations are simplified by separating the turbulent flow into a mean (ui, p) and a fluctuating part (u ′ i, p ′) and restricting the analysis to time- averages of the turbulent motion. ui = ui + u ′ i, p = p + p ′, (C.3) 129 130 Appendix C. Numerical flow modeling This is called Reynolds decomposition. The mean quantities are defined as ui = 1 T ∫ T 0 uidt, p = 1 T ∫ T 0 pdt (C.4) where the averaging time T must be sufficiently large (compared with the time scale of the turbulent motion) for the average value to approach the real time- independent mean value. Substituting Eq. (C.3) into Eqs. (C.2) and (C.1) and subsequent averaging leads to a system of equations for the mean motion. For brevity, the overbars indicating averaged values will be dropped from ui, and p from here on. The mean continuity equation is as follows ∂ui ∂xi = 0 (C.5) And the mean momentum equations are ∂ui ∂t + uj ∂ui ∂xj = −1 ρ ∂p ∂xi + ν ∂2ui ∂x2j − ∂u′iu ′ j ∂xj + fi (C.6) The −u′iu′j terms represent the contribution of the turbulent motion to the mean stress. The turbulent stresses −ρu′iu′j are called the Reynolds stresses. The process of averaging has introduced unknown terms representing the transport of mean momentum by turbulent motion. Consequently, this set of equations cannot be solved without additional information. This is known as the closure problem of turbulence. It has led to the development of turbulence models, in which the Reynolds stresses are modeled. An extensive review of turbulence models and their application in hydraulics can be found in Rodi (1993). An assessment of possible turbulence modelings that can be used for the design of bed protections is discussed in Hofland (2005, chapter 8). In this research the two-equation k − ε model was chosen as it is widely tested and used for hydraulic flow problems. The k − ε model employs conservation equations for the rate of turbulent kinetic energy k and for the rate of energy dissipation ε. In the next section, the k-ε model will be described in somewhat more detail. C.1.2 The two-equation k-ε model In the two-equation k− ε model, two extra transport equations are introduced to represent the turbulent properties of the flow. For turbulent kinetic energy: ∂k ∂t + ∂ ∂xi (kui) = ∂ ∂xi [( ν + νt σk ) ∂k ∂xi ] + Pk − ε (C.7) C.1. Turbulence modeling 131 For turbulent dissipation: ∂ε ∂t + ∂ ∂xi (εui) = ∂ ∂xi [( ν + νt σε ) ∂ε ∂xi ] + c1ε ε k Pk − c2ε ε 2 k (C.8) where ε is the turbulent dissipation that determines the scale of the turbulence, k is the turbulent kinetic energy that determines the energy in the turbulence, Pk is the production rate of turbulent energy given by: Pk = −u′iu′j ∂ui ∂xj (C.9) The eddy viscosity is modeled as: νt = cµ k2 ε (C.10) The model contains some closure constants which are given as follows: cµ = 0.09, c1ε = 1.44, c2ε = 1.92, σk = 1.0, σε = 1.3 (C.11) With the use of the constants in Eq. (C.11) we have the standard k− ε model. In the standard k − ε model the eddy viscosity is determined from a single tur- bulence length scale, so the calculated turbulent diffusion is that which occurs only at the specified scale. In reality all scales of motion will contribute to the turbulent diffusion. A mathematical technique used to account for the different scales of motion through changes to the production term results in the so-called RNG k − ε model (Yakhot et al., 1992). The RNG k − ε model is based on Re- Normalisation Group (RNG) analysis of the Navier-Stokes equations, to account for the effects of smaller scales of motion. The transport equations for turbulence generation and dissipation are the same as those for the standard k− ε model but the model constants are different and are given as follows (Segal et al., 2000): cµ = 0.085, c1ε = 1.42− η(1− η/η0) 1 + γη3 , c2ε = 1.68, σk = σε = 0.7179 (C.12) with η0 = 4.38, γ = 0.012 and η = Sk/ε. S is the magnitude of the mean rate of strain, defined as S = (2sijsij) 1 2 . sij is the mean rate of strain: sij = 1 2 ( ∂ui ∂xj + ∂uj ∂xi ) (C.13) 132 Appendix C. Numerical flow modeling C.2 Deft input files The turbulent flows through the flume have been simulated using Deft incom- pressible flow solver. In this section a complete description of the typical Deft input files used in our simulations is presented. Three following file types are used as Deft input files: *.msh - grid generation, *.prb - problem description and *.f - file contains function subroutine USFUNB used to customize the boundary conditions such as inlet velocity and turbulence distributions. The first stage of an Deft job is to generate a grid. The geometry of the flume set-up leads to the choice of multi block approach. The grid information is de- scribed in a text file (*.msh) with special format that SEPRAN grid generator SEPMESH can understand. In the mesh file, points, curves, surfaces and vol- umes are defined. The boundaries are also marked so that they can be prescribed in *.prb file (see Figure 6.1). The content of a typical mesh file (*.msh) is de- scribed in Section C.2.1. The next stage of a Deft job is to specify the physical, mathematical and solu- tion parameters of the problems. This includes for example the specification of the viscosity, the density, the boundary conditions, the turbulence model. These are presented and discussed in Section C.2.2. The description of the sequence of a Deft session is presented in Section C.2.3. C.2.1 Mesh description The 37 flow conditions (see Table 6.1) that were modeled requires 37 different mesh files. The following input file 1ar.msh for SEPMESH was used to generate the grid for flow condition 1AR . To make it easy to use for other flow conditions, the main dimensions of the model set-up are defined in the mesh file as constants and can be changed to proper values for each simulation. The plot command at the end of the file is used to generate the graphic output files. These files are used to visually check the grid generation. * 3D mesh for an open-channel with gradual expansion * The constants below are applied for all flow conditions * in set-up 1 except the water depth h * Dimensions are in meters. constants reals x1 = 7.6 # start of the expansion x2 = 10.5 # end of the expansion x3 = 11.5 # flume length y1 = 0.075 # distance between inside and outside walls y2 = 0.25 # half of the flume width h = 0.12 # water depth C.2. Deft input files 133 integers nx1 = 130 # the number of cells in the first straight # part of the flume in x-direction (7.6 m) nx2 = 100 # the number of cells in the expansion # part in x-direction (2.9 m) nx3 = 35 # the number of cells in the second straight # part of the flume in x-direction (1.0 m) ny = 8 # the number of cells in y-direction nz = 10 # the number of cells in z-direction end mesh3d isnas points p1=(0,$y1,0) p2=($x1,$y1,0) p3=($x2,0,0) p4=($x3,0,0) p5=($x3,$y2,0) p6=($x2,$y2,0) p7=($x1,$y2,0) p8=(0,$y2,0) p9=(0,$y1,$h) p10=($x1,$y1,$h) p11=($x2,0,$h) p12=($x3,0,$h) p13=($x3,$y2,$h) p14=($x2,$y2,$h) p15=($x1,$y2,$h) p16=(0,$y2,$h) curves c1=line1(p1,p2,nelm=$nx1, ratio = 1, factor = 0.25) c2=line1(p2,p3,nelm=$nx2) c3=line1(p3,p4,nelm=$nx3) c4=line1(p4,p5,nelm=$ny) c5=line1(p5,p6,nelm=$nx3) c6=line1(p6,p7,nelm=$nx2) c7=line1(p7,p8,nelm=$nx1, ratio = 1, factor = 4) c8=line1(p8,p1,nelm=$ny) c9=line1(p1,p9,nelm=$nz, ratio = 1, factor = 2) c10=translate c9(p2,p10) c11=translate c9(p3,p11) c12=translate c9(p4,p12) c13=translate c9(p5,p13) 134 Appendix C. Numerical flow modeling c14=translate c9(p6,p14) c15=translate c9(p7,p15) c16=translate c9(p8,p16) c17=translate c8(p7,p2) # x-y plan at z = 0 c18=line1(p6,p3,nelm=$ny) c19=translate c1(p9,p10) # x-y plan at z = h c20=translate c2(p10,p11) c21=translate c3(p11,p12) c22=translate c4(p12,p13) c23=translate c5(p13,p14) c24=translate c6(p14,p15) c25=translate c7(p15,p16) c26=translate c8(p16,p9) c27=translate c26(p15,p10) # x-y plan at z = h c28=line1(p14,p11,nelm=$ny) surfaces s1=rectangle5(c1,-c17,c7,c8) # flume bottom s2=rectangle5(c2,-c18,c6,c17) # flume bottom s3=rectangle5(c3,c4,c5,c18) # flume bottom s4=rectangle5(c1,c10,-c19,-c9) # flume wall s5=rectangle5(c2,c11,-c20,-c10) # flume wall s6=rectangle5(c3,c12,-c21,-c11) # flume wall s7=rectangle5(c4,c13,-c22,-c12) # outflow s8=rectangle5(-c7,c15,c25,-c16) # symetric wall s9=rectangle5(-c6,c14,c24,-c15) # symetric wall s10=rectangle5(-c5,c13,c23,-c14) # symetric wall s11=rectangle5(-c8,c16,c26,-c9) # inflow s12=rectangle5(c19,-c27,c25,c26) # surface s13=rectangle5(c20,-c28,c24,c27) # surface s14=rectangle5(c21,c22,c23,c28) # surface s15=rectangle5(-c17,c15,c27,-c10) s16=rectangle5(-c18,c14,c28,-c11) volumes v1=brick13(s1,s4,s15,s8,s11,s12) v2=brick13(s2,s5,s16,s9,s15,s13) v3=brick13(s3,s6,s7,s10,s16,s14) plot end The values of the above constants (i.e., x1, x2, x3, y1, y2) are applied for all flow conditions in set-up 1 (i.e., 1AR to 1LR). The water depth h varies from 0.12 m to 0.19 m depending on flow conditions. For the flows in set-up 2 and 3, the following constants are used. Again, the water depth h varies accordingly. C.2. Deft input files 135 * The constants below are applied for all flow conditions * in set-up 2 except the water depth h * Dimensions are in meters. constants reals x1 = 8.2 # start of the expansion x2 = 9.9 # end of the expansion x3 = 11.5 # flume length y1 = 0.075 # distance between inside and outside walls y2 = 0.25 # half of the flume width h = 0.12 # water depth integers nx1 = 130 # the number of cells in the first # straight part of the flume in x-direction nx2 = 60 # the number of cells in the expansion # length in x-direction nx3 = 40 # the number of cells in the second straight # part of the flume in x-direction ny = 8 # the number of cells in y-direction nz = 10 # the number of cells in z-direction end * The constants below are applied for all flow conditions * in set-up 3 except the water depth h * Dimensions are in meters. constants reals x1 = 8.2 # start of the expansion x2 = 9.4 # end of the expansion x3 = 11.5 # flume length y1 = 0.075 # distance between inside and outside walls y2 = 0.25 # half of the flume width h = 0.12 # water depth integers nx1 = 130 # the number of cells in the first # straight part of the flume in x-direction nx2 = 45 # the number of cells in the expansion # length in x-direction nx3 = 50 # the number of cells in the second straight # part of the flume in x-direction ny = 8 # the number of cells in y-direction nz = 10 # the number of cells in z-direction end 136 Appendix C. Numerical flow modeling C.2.2 Problem description In Deft incompressible flow solver the problem description is specified in a text file (*.prb). The Deft pre-processor (ISNASPRE) read this input file and inter- prets it. The same description is used for all the 37 flow simulations and is listed below. The only difference among various simulations (flow conditions) is the definition of the velocity distribution at the inlet. This is treated in a separate *.f file. * *3D turbulent flow through an open-channel with gradual expansion * turbulence model = k_eps kappa = 0.4187 E = 9.793 discretization turbulence_equations all upwind = first_order time_integration tinit = 0 tend = 30 tstep = 0.008 theta = 1 rel_stationary_accuracy = 1d-2 boundary_conditions * bottom curve 1 to 3: wall_functions = roughness roughness = 0.02 * inflow curve 11: un = func = 1, ut1 = 0, ut2 = 0 k_dirichlet = 1.5d-3 eps_dirichlet = 8.624833d-5 * outflow curve 7: outflow k_neumann = 0 eps_neumann = 0 * side wall curve 4 to 6: wall_functions = roughness roughness = 0.005 C.2. Deft input files 137 * symmetry curve 8 to 10: freeslip k_neumann = 0 eps_neumann = 0 * top curve 12 to 14: freeslip k_dirichlet = 9.0d-4 eps_dirichlet = 8.624833d-5 coefficients momentum_equations rho = 1d3 mu = 1d-3 force3 = 10 multi_block subdomain_solution = inaccurate linear_solver momentum_equations amount_of_output = 0 relaccuracy = 1d-3 maxiter = 1000 pressure_equations amount_of_output = 0 divaccuracy = 0 relaccuracy = 1d-4 startvector = zero maxiter = 1000 turbulence_equations all amount_of_output = 0 relaccuracy = 1d-3 maxiter = 1000 To describe the boundary condition at the inlet, the function subroutine USFUNB is used. The following file 1ar.f is used for the flow condition 1AR. For other flow conditions, the corresponding values of water depth (h) and discharge (Q) are applied. program isnasexe implicit none integer nbuffr parameter( nbuffr = 300000000 ) 138 Appendix C. Numerical flow modeling integer ibuffr common ibuffr(nbuffr) call ishmain( nbuffr ) end function usfunb ( ichoice, x, y, z, t ) c User written function subroutine. It gives c the user the opportunity to define a c boundary condition as a function of space c and time. implicit none double precision usfunb, x, y, z, t integer ichoice c x i x-coordinate c y i y-coordinate c t i actual time c ichoice i choice parameter given by the user input c usfunb o computed boundary condition double precision Q, B, h, u c Q discharge c B channel width c h water depth c u inflow velocity u = u(z) Q = 22d-3 B = 35d-2 h = 12d-2 u = 3*Q*z*z/(B*h*h*h) if ( ichoice.eq.1 ) then usfunb = -u end if end C.2. Deft input files 139 C.2.3 Typical sequence of an Deft session Once the Deft input files have been created, certain commands must be given to run the simulation. Following are typical commands used to simulate a flow condition, for instance, 1AR. sepmesh 1ar.msh # grid generation isnaspre 1ar.prb # read and interpret problem description islink 1ar # submit the function subroutine USFUNB qsub runisnas # run the simulation 140 Appendix C. Numerical flow modeling List of Symbols Roman Symbols a coefficient - A area m2 b coefficient - B width (of flume) m C Chezy coefficient √ m/s ciε, cµ closure constants in k− ε turbulence model (i = 1, 2) - d stone, particle diameter m dn nominal stone diameter (≡ 3 √ V) m dn50 median nominal diameter( ≡ 3 √ m50/ρs) m dx stone diameter where x%of the stonemass has a smaller diameter m E entrainment rate - Em measured entrainment rate, without correction - Ec corrected entrainment rate - f weighing function (≡ (1− z/H)β) - F flow force N F1 friction force N F2 resistance force N FD drag force N Fg gravitational force N FL lift force N Fmax (estimate of) maximum (extreme) occurring force N Fr Froude number (≡ U/√gh) - g gravitational acceleration m/s2 h water depth m H water column height above the bed m k turbulence kinetic energy m2/s2 Kh water depth parameter - ks equivalent roughness m Ks slope correction factor - 141 142 List of Symbols ksb bottom roughness m ksw side wall roughness m KT turbulence correction factor - Kv velocity/turbulence correction factor - l stone displacement length m L stone strip width m L˜ dimensionless strip width (≡ L/l¯) - lm mixing length m Lm Bakhmetev mixing length m m mass kg n number of displaced stones - p pressure N/m2 Pk production rate of turbulent energy m 2/s3 qs bed load transport per m width m 2/s Q discharge m3/s R hydraulic radius (≡ ω/χ) m R2 coefficient of determination - Re Reynolds number (≡ Uh/ν) - Re∗ particle Reynolds number (≡ u∗dn/ν) - S empirical factor accounts for the way the stones are placed - sij mean rate of strain s −1 t time s T period, time-scale or duration s u streamwise velocity m/s u∗ shear velocity (≡ √ τb/ρ) m/s u∗c critical shear velocity m/s ub near bed streamwise velocity m/s uc,u critical flow velocity in uniform flow m/s uc,nu critical flow velocity in non-uniform flow m/s umax maximum streamwise velocity m/s U cross-sectional average of streamwise velocity m/s v transverse velocity m/s V volume (of stone) m3 w upward velocity m/s x coordinate in direction of flow m y transverse coordinate m z vertical coordinate m z0 roughness length m Greek symbols α empirical constant (various uses) or - expansion angle degree β empirical constant (various uses) or - Clauser’s parameter - List of Symbols 143 δ boundary layer thickness m δ∗ displacement thickness m ∆ specific submerged density of stone (≡ ρs/ρ − 1)) - ε turbulence dissipation κ Von Karman constant - ν kinematic viscosity m2/s νt eddy viscosity m 2/s Π Coles wake parameter - ρ density of water kg/m3 ρs density of stone, epoxy resin, polyfit or sand kg/m 3 σk, σε closure constants in k− ε turbulence model - τ shear stress N/m2 τb bed shear stress N/m 2 τc critical bed shear stress N/m 2 Φ transport parameter (bed damage indicator) - ΦE entrainment parameter (dimensionless entrainment rate) - Φq dimensionless bed load transport - Ψ stability parameter (ratio of load to strength) - Ψc critical stability parameter - ΨLm Hofland stability parameter - Ψu−σ[u] stability parameter using u and σ[u] - Ψs Shields stability parameter (≡ τb/∆gd) - Ψs,c critical Shields stability parameter - ΨWL stability parameter developed at WL|Delft Hydraulics - ΨWL,c critical value of ΨWL - ω wetted cross-sectional area (≡ B× h) m2 χ wetted perimeter (≡ B + 2h) m Mathematics |x| absolute value of x x temporal average of x 〈x〉 spatial average of x x′ fluctuating part of x around x xˆ predicted value of x x˜ dimensionless form ≡ defined as ∝ proportional to ≈ approximately equal to D D material derivative ∂ ∂ partial derivative∫ integral 144 List of Symbols ± plus or minus f (x) unspecified function of x δx relative difference between two values of x ∆x difference between two values of x Abbreviations 2D two-dimensional 3D three-dimensional BFS backward-facing step EMS Electro Magnetic velocity Sensor LDV Laser Doppler Velocimeter QSF quasi-steady forces RANS Reynolds-averaged Navier-Stokes TWP turbulence wall pressures List of Figures 1.1 Graphical presentation of the researchmethodology and thesis lay- out. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Turbulence intensity distributions downstream of a sudden expan- sion after El-Shewey and Joshi (1996). . . . . . . . . . . . . . . . . . 13 2.2 Forces acting on particles resting on a bed surface . . . . . . . . . . 14 2.3 Four mechanical model concepts after Mosselman et al. (2000). . . . 22 2.4 Original Shields curve (1936). The hatched area shows the critical shear stress as a function of the particle Reynolds number. . . . . . 23 2.5 Variation of the dimensionless bed load transport with the Shield stability parameter after Paintal (1971) (large particles only). . . . . 27 2.6 Top: measured ΦE versus the measured ΨWL for a variety of flow conditions. Bottom: measured ΦE versus the measured ΨLm for the same flow conditions with the tentative curve expressed by Eq. (2.46) (Hofland, 2005). . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 A summary of stone stability assessment methods. . . . . . . . . . . 29 3.1 Experimental installation (not to scale). . . . . . . . . . . . . . . . . 34 3.2 The first experimental configuration indicating the placement of uniformly colored artificial stone strips (not to scale). . . . . . . . . 35 3.3 Longitudinal sections of the three experimental configurations. . . 35 3.4 Distributions of velocity and Reynolds shear stress. . . . . . . . . . 36 3.5 Horizontal distributions of mean velocity (u and v) at the middle cross section of the expansion. The measurement was undertaken for flow condition B (top panel) and L (bottom panel). . . . . . . . . 37 3.6 The relative errors (δ) of 2-minute (square), 5-minute (plus) and 10-minute (dot) sub-signals to the 30-minute signal. The measure- ments were carried out at profile 1 under flow condition 1FR. . . . 44 4.1 Typical velocity profiles. . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Distributions of eddy viscosity and mixing length (set-up 1). . . . . 56 4.3 Distributions of eddy viscosity and mixing length (set-up 2). . . . . 57 4.4 Distributions of eddy viscosity and mixing length (set-up 3). . . . . 58 4.5 Turbulence intensity distributions (set-up 1). . . . . . . . . . . . . . 60 145 146 List of Figures 4.6 Turbulence intensity distributions (set-up 2). . . . . . . . . . . . . . 61 4.7 Turbulence intensity distributions (set-up 3). . . . . . . . . . . . . . 62 4.8 Reynolds shear stress distributions. . . . . . . . . . . . . . . . . . . . 64 5.1 The distributions of key parameters used to formulate the new stability parameter. From left to right: extreme force distribution (a), weighting function (b) and weighting average of the extreme forces (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Sensitivity analysis of α, β and H. . . . . . . . . . . . . . . . . . . . . 70 5.3 Vertical distributions of key parameters in Eq. (5.7). . . . . . . . . . 71 5.4 Measured Ψu−σ[u] versus measured ΦE. . . . . . . . . . . . . . . . . 71 5.5 Measured Ψs versus measured ΦE. . . . . . . . . . . . . . . . . . . . 73 5.6 Measured ΨWL (with α = 6) versus measured ΦE. . . . . . . . . . . 74 5.7 Sensitivity analysis of α in Eqs. (2.27) and (2.28). . . . . . . . . . . . 74 5.8 Measured ΨWL (with α = 3.5) versus measured ΦE. . . . . . . . . . 75 5.9 Measured ΨLm (with α = 6) versus measured ΦE. . . . . . . . . . . 76 5.10 Measured ΨLm (with α = 3) versus measured ΦE. . . . . . . . . . . 77 5.11 Typical distributions of the key parameters according to Eqs. (2.26), (2.27), (2.28) and (5.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.12 Sensitivity analysis of the velocity and turbulence to the bed dam- age. The α values of 3.5, 3 and 3 are used in Eqs. (2.27), (2.28) and (5.7), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.13 Data comparison. The α value of 6 was used for both ΨWL (top) and ΨLm (bottom). The comparison was made for the measured entrainment data, i.e. ΦE ≡ ΦEm. . . . . . . . . . . . . . . . . . . . . 82 5.14 Data comparison. The α value of 6 was used for both ΨWL (top) and ΨLm (bottom). The comparison was made for the corrected entrainment data, i.e. ΦE ≡ ΦEc. . . . . . . . . . . . . . . . . . . . . 83 6.1 Definition region of the model set-up. . . . . . . . . . . . . . . . . . 90 6.2 Grid refinement test (velocity and turbulence). . . . . . . . . . . . . 90 6.3 Profiles of calculated (lines) and measured (circles) flow parame- ters over the flume with flow condition 1AR. . . . . . . . . . . . . . 93 6.4 Comparison between the standard k− ε and RNG k− ε model. . . 95 6.5 Comparison of calculations (lines) andmeasurements (circles). From top to bottom: flow condition 1BR, 2BR, 3BR, 1LR, 2LR and 3LR. The results at profile 1 to 4 are plotted from left to right. . . . . . . . 96 6.6 Vertical distributions of key parameters in Eqs. (2.27) and (2.28). The turbulence magnification factor α = 3.5 was used. Left: flow condition 2BR. Right: flow condition 2IR. . . . . . . . . . . . . . . . 98 6.7 Comparison of measured and calculated stability parameters. . . . 98 List of Figures 147 6.8 Comparison of measured and calculated bed damage (ΦE). The calculated bed damage was determined using Eq. (6.6) [left] and Eq. (6.7) [right]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 A.1 The grading curves of artificial (line) and natural (dash) stones. . . 116 B.1 Calculations (lines) and measurements (circles) of streamwise ve- locity (u) and turbulence intensity (k). From top to bottom: flow condition 1AR, 1BR, 1CR, 1DR, 1ER and 1FR. The results at profile 1 to 4 are plotted from left to right. . . . . . . . . . . . . . . . . . . . 118 B.2 Calculations (lines) and measurements (circles) of streamwise ve- locity (u) and turbulence intensity (k). From top to bottom: flow condition 1GR, 1HR, 1IR, 1JR, 1KR and 1LR. The results at profile 1 to 4 are plotted from left to right. . . . . . . . . . . . . . . . . . . . 119 B.3 Calculations (lines) and measurements (circles) of streamwise ve- locity (u) and turbulence intensity (k). From top to bottom: flow condition 2AR, 2BR, 2CR, 2DR, 2ER and 2FR. The results at profile 1 to 4 are plotted from left to right. . . . . . . . . . . . . . . . . . . . 120 B.4 Calculations (lines) and measurements (circles) of streamwise ve- locity (u) and turbulence intensity (k). From top to bottom: flow condition 2GR, 2HR, 2IR, 2JR, 2KR and 2LR. The results at profile 1 to 4 are plotted from left to right. . . . . . . . . . . . . . . . . . . . 121 B.5 Calculations (lines) and measurements (circles) of streamwise ve- locity (u) and turbulence intensity (k). From top to bottom: flow condition 3AR, 3BR, 3CR, 3DR, 3ER and 3FR. The results at profile 1 to 3 are plotted from left to right. . . . . . . . . . . . . . . . . . . . 122 B.6 Calculations (lines) and measurements (circles) of streamwise ve- locity (u) and turbulence intensity (k). From top to bottom: flow condition 3GR, 3HR, 3IR, 3JR, 3KR and 3LR. The results at profile 1 to 3 are plotted from left to right. . . . . . . . . . . . . . . . . . . . 123 B.7 Calculations (lines) and measurements (circles) of streamwise ve- locity (u) and turbulence intensity (k) of flow condition 3MR. The results at profile 1 to 3 are plotted from left to right. . . . . . . . . . 124 148 List of Figures List of Tables 2.1 List of dominant governing variables. . . . . . . . . . . . . . . . . . 16 3.1 The main characteristics of stones used in the experiments. . . . . . 39 3.2 A possible set of hydraulic conditions. . . . . . . . . . . . . . . . . . 40 3.3 Summary of hydraulic conditions measured from the experiments. 42 3.4 Summary of measurements and equipments used in the experiment. 43 4.1 Summary of measured and calculated flow parameters (set-up 1). . 50 4.2 Summary of measured and calculated flow parameters (set-up 2). . 51 4.3 Summary of measured and calculated flow parameters (set-up 3). . 52 4.4 The empirical constants α and β determined from the present data. 59 5.1 Coefficient of determination for different data sets. . . . . . . . . . . 72 6.1 The flow conditions that are to be modeled. . . . . . . . . . . . . . . 88 6.2 Grid refinement test for set-up 2. . . . . . . . . . . . . . . . . . . . . 89 6.3 Grid refinement for the three flume set-ups. . . . . . . . . . . . . . . 91 B.1 Summary of measured governing variables (set-up 1, α = 30). . . . 125 B.2 Summary of measured governing variables (set-up 2, α = 50). . . . 126 B.3 Summary of measured governing variables (set-up 3, α = 70). . . . 127 149 150 List of Tables Acknowledgements This thesis would have not been realized without the support and contributions of many people. Therefore, I would like to thank all of them, including those I may have forgotten to mention here. First of all, I would like to thank all the members of my guidance committee, Marcel Stive, Rob Booij, Bas Hofland, and Henk Jan Verhagen, for their thought- ful guidance and encouragement during this study. Thank you for allowing me freedom in doing my research. This is especially true when you want to switch the research focus in the middle of the second year of your Ph.D. study as I did. I especiallywant to thank Rob Booij and BasHofland,my daily supervisors, for the good guidance and incisive advice. Thank you very much for sharing your great knowledge of ”Rock & Roll”and for always reviewing my manuscripts. I thank Henk Jan Verhagen for recommending me to follow this research topic from the days I was still in Vietnam and for his active supervision during all stages of this research. And I thank Marcel Stive for being my great promotor. Your construc- tive criticism and comments, especially at the early stage of writing this thesis, have motivated me to improve my writing. Thank you very much for your help, support and for correcting my thesis. I would like to thank all the staff members of the Laboratory of Fluid Mechan- ics of Delft University of Technology for their support duringmy one-year period working in the lab. I would have been lost during the measurements without Wim Uijttewaal and Sander de Vree. Your guidance, enthusiasm, valuable ad- vice and assistance are much appreciated. Jaap van Duin and Ben Lemmers are thanked for setting up the experiments. The artificial stones would have not been made without the help of Arie den Toom. Thank you very much. I thank Marcel Zijlema for his valuable advice and assistance in the flow sim- ulations using Deft incompressible flow solver. I would have not been able to perform the simulations without you. Many thanks go to Rosh Ranasinghe for his time and effort in reviewing chapter 3 and 5 of this thesis. I would like to thank all the members of my promotion committee for spend- ing time and effort to review this thesis and provide me with insightful remarks. I would like to thank all the staff members of CICAT, especially Veronique van der Varst and Paul Althuis, for their support and help. My four years in Delft would have been much more difficult without your assistance. 151 152 Acknowledgements This research was funded by the Ministry of Education and Training of Viet- nam, CICAT and the Hydraulic Engineering Department of Delft University of Technology. This financial support is gratefully acknowledged. Also the finan- cial support from the Lamminga Foundation to return Delft for the defense of this thesis is much appreciated. I thank all the members of the Hydraulic Engineering Department, especially Kees den Heijer, Tomohiro Suzuki, Fred Vaes and my Vietnamese colleagues, for creating a wonderful working atmosphere. I feel lucky to have fun roommates like Kees den Heijer and Nghiem Tien Lam, who could always understand all those Vietnamese-style jokes. Thank you for the joyful atmosphere and exchange of remarks and thoughts. Kees, thank you for taking care of all the formalities while I was in Vietnam and for translating the summary into Dutch. Many thanks also go to Mark Voorendt, Chantal van Woggelum, Inge van Rooij and Adeeba Ramdjan for all their help and support. Special thanks go to all my Vietnamese friends for making my four-year stay in Delft very pleasant. I will not forget the football matches every Saturday, and the exercises (and jokes) at the gym with Nguyen Dai Viet. I was lucky enough to stay in the same nice house in Delft for the last four years thanks to brother Vu Ngoc Pi and Nguyen Tien Hung. I will never forget the nice time we had. I would like to thank my colleagues at the Port and Waterway Engineering Department, Hanoi University of Civil Engineering (HUCE) for their support. I especially want to thank Prof.dr. Luong Phuong Hau for his guidance, support and encouragement since the days I joined HUCE as a lecturer. I have learned a lot more than hydraulic engineering from you. Ass.prof.dr. Pham Van Giap is thanked for recommending me to study hydraulic engineering. And I thank Ms. Dao Tang Kiem for helping me to apply for the Ph.D. position at Delft University of Technology. A nature word of thanks goes to my parents. Their love, support and trust have encouraged me to pursue my personal objectives. I would like to thank my brother Nguyen Thuan andmy sister in-lawNguyenMinhQuy for taking care of our parents and for being my great brother and sister. A sincere appreciation to my parents in-law for their love, support and for taking care of my little family while I was away. I could not have focused on the study without their help and support. I also want to take this opportunity to thank my uncle Phung Xuan Minh andmy aunt Nguyen Thi Nhamwho have cared aboutme and have always supported me. Thank you very much for your support and trust. Last, but certainly not least, I would like to express my deepest gratitude to my beloved wife Hai Ly, who has unconditionally supported me. It would have been impossible to reach this day without your love, support, motivation and patience. Thank you for being my wonderful wife. Together with our daughter Tra My, you fill my life with happiness and inspiration every day. Curriculum Vitae Nguyen ThanhHoan was born on 24 September 1976 in Nam Dinh, Vietnam. He obtained his high school diploma from Le Hong Phong (Nam Dinh) secondary school in 1994. During the period at secondary school he won a third prize of the National Talent Competition on Physics Subject (thi hoc sinh gioi quoc gia mon Vat Ly, 1993). In 1994 Hoan started his university in Hanoi University of Civil Engineering (HUCE), where he obtained BSc degree, with distinction (one of the three of the university that year), in hydraulic engineering in March 1999. During the period at university he won two third prizes of the National Student Olympic Competition - a prime annual academic event amongst technical universities in Vietnam - on The- oretical Mechanics Subject (1996) and on Structural Mechanics Subject (1997, this one without the first and second prizes). Shortly after graduation, in June 1999 he was employed as an assistant lecturer at the Faculty of Hydraulic Engineer- ing, HUCE while continuing his master study at the Faculty of Postgraduates, HUCE. He obtained his Master degree in 2002 on a thesis entitled ”Application of physical model in investigating river revolution”. In July 2004, he started his Ph.D. study at the Department of Hydraulic Engineering, the Faculty of Civil Engineering and Geosciences, Delft University of Technology which led to this thesis. Currently Hoan works as a lecturer at the Department of Ports and Wa- terway Engineering, Hanoi University of Civil Engineering. 153 154 Curriculum Vitae Propositions Pertaining to the thesis Stone Stability under Non-uniform Flow by Nguyen Thanh Hoan Delft, 3 November 2008 1. Assessing the hydraulic loads exerted on the stones on a bed and the associated stability of the stones are central in stone stability research. 2. Although near-bed velocities cause the main forces on bed material, flow parame- ters at different depths can be used to represent the hydraulic loads exerted on the bed. (Hofland, 2005 and this thesis) Hofland, B. (2005). Rock & roll: turbulence-induced damage to granular bed protections. Ph.D. thesis, Delft University of Technology. 3. A combination of velocity and turbulence distributions should be used to quantify the hydraulic loads exerted on the stones on a bed. (this thesis) 4. The popular stability threshold concept should be used with care; it may yield inconsistent and unreliable design criteria. The stability transport concept may overcome this. 5. With the availability of the newly-developed stone transport formulae and more reliable turbulencemodels, the bed damage level can be more accurately computed for arbitrary flow conditions. (this thesis) 6. Beneath a good formula for bed processes is a good physical bedding. 7. Doing a Ph.D. study abroad may be very stressful but also very enjoyable. 8. For a starting Ph.D. student, a lack of background knowledge on the subject may be an advantage. 9. A good scientist and a good flume are necessary but not sufficient conditions to perform a good experiment. 10. Strongly economically developing countries such as Vietnam, need to develop a home-base for scientific research. These propositions are considered opposable and defendable, and as such have been approved by the supervisor, Prof.dr.ir. M.J.F. Stive Stellingen Behorende bij het proefschrift Steenstabiliteit onder Niet-uniforme Stroming van Nguyen Thanh Hoan Delft, 3 november 2008 1. Bepaling van de hydraulische belastingen op de stenen op de bodem en de daaraan gerelateerde stabiliteit van de stenen staan centraal in steenstabiliteitsonderzoek. 2. Hoewel de stroomsnelhedennabij de bodemde belangrijkste krachten op het bodem- materiaal veroorzaken, kunnen stromingsparameters op verschillende diepten ge- bruikt worden omde hydraulische krachten op de bodem te representeren. (Hofland, 2005 en dit proefschrift) Hofland, B. (2005). Rock & roll: turbulence-induced damage to granular bed protections. Ph.D. thesis, Delft University of Technology. 3. Een combinatie van stroomsnelheid en turbulentieverdelingen zou gebruiktmoeten worden om de hydraulische belastingen op de stenen op een bodem te kwantifi- ceren. (dit proefschrift) 4. Voorzichtigheid is geboden bij gebruik van het populaire concept voor de drem- pelwaarde van de stabiliteit; het kan leiden tot inconsistente en onbetrouwbare ontwerpcriteria. Het stabiliteitstransport concept kan hierin uitkomst bieden. 5. Met de beschikbaarheid van de nieuw-ontwikkelde steentransportformules en be- trouwbaardere turbulentiemodellen kan het bodemschadeniveau preciezer wor- den berekend voor willekeurige stromingscondities. (dit proefschrift) 6. Aan een goede formule voor bodemprocessen moet een goede fysische bedding ten grondslag liggen. 7. Een promotieonderzoek in het buitenland doen, kan leiden tot veel stress, maar ook veel plezier. 8. Een gebrek aan achtergrondkennis over het onderwerp kan een voordeel zijn voor een beginnende promovendus. 9. Een goedewetenschapper en een goede goot zijn noodzakelijkmaar niet voldoende om een goed experiment uit te kunnen voeren. 10. Sterk economisch onwikkelende landen zoals Vietnam, hebben een thuisbasis voor wetenschappelijk onderzoek nodig. Deze stellingenworden opponeerbaar en verdedigbaar geacht en zijn als zodanig goedge- keurd door de promotor, Prof.dr.ir. M.J.F. Stive