Pier and Contraction Scour in Cohesive Soils (2004)

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45 CHAPTER 6 THE SRICOS-EFA METHOD FOR INITIAL SCOUR RATE AT COMPLEX PIERS 6.1 GENERAL The initial scour rate is an integral part of the SRICOS Method because it is one of the two fundamental parameters used to describe the scour depth versus time curve. The other fundamental parameter is the maximum depth of scour, which was studied in Chapter 5. The initial rate of scour for a given complex pier scour problem is obtained by first calculating the maximum shear stress τmax existing around the pier before the scour hole develops (flat river bottom) and then reading the initial scour rate on the erosion function obtained in the EFA test. Therefore, the problem of obtaining the initial rate of scour is brought back to the problem of obtaining the maxi- mum shear stress around the pier before scour starts. This problem was solved by using numerical simulations. The sim- ulations performed and the associated results are described in this chapter. The goal was to develop correction factors for giving τmax for a cylindrical pier in deep water (Equation 6.1): These factors include the effects of shallow water depth, pier shape, pier spacing, and angle of attack. 6.2 EXISTING KNOWLEDGE ON NUMERICAL SIMULATIONS FOR SCOUR Hoffman and Booij (1993) applied the Duct Model and the Sustra Model to simulate the development of local scour holes behind the structure. The flow model upon which Duct is based is a parabolic boundary-layer technique using the finite element method. The Sustra Model is used to compute the concentration field following the approach used by Van Rijn and Meijer (1986). The computational model results were compared with experimental data. The results (i.e., flow velocities, sediment concentration, and bed configurations as a function of time) showed an agreement between the exper- imental data and the computational model. Olsen and Melaaen (1993) simulated scour around a cylin- der by using SSIIM, a three-dimensional free-surface flow and transport model. The SSIIM Model solves Reynold stresses by the k −
turbulence model. The authors observed and reported that there is agreement between the pattern of the vortices in τ ρmax . log ( . )= −  0 094 1 110 6 12v Re front of the cylinder and the model. The application of the SSIIM Model can be found at http://www.sintef.no/nhl/vass/ vassdrag.html. Wei et al. (1997) performed a numerical simulation of the scour process in cohesive soils around cylindrical bridge piers. A multiblock chimera Reynolds-averaged Navier Stokes (RANS) method was incorporated with a scour rate equation to simulate the scour processes. The scour rate equation linked the scour rate to the streambed shear stress through a linear function. The simulation captured the important flow features such as the horseshoe vortex ahead of the pier and the flow recirculation behind the pier. A reasonable agreement was found between the progress of the scour depth obtained in the flume experiments and predicted by the numerical simulation. Wei et al. found that the value of the critical shear stress has a significant influence on the scour process around a cylinder in cohesive soils. The final scour depth and the time necessary to reach it increase with decreasing critical shear stress. Based on a number of parametric runs, they also presented an empirical formula for the maximum streambed shear stress for a cylin- drical pier in deep water. Dou (1997) simulated the development of scour holes around piers and abutments at bridge crossings. A stochastic turbulence closure model (Dou, 1980), which includes an iso- tropic turbulence, was incorporated into a three-dimensional flow model, CCHE3D, developed by the Center for Computa- tional Hydroscience and Engineering at the University of Mississippi. The factors that reflect the secondary flow motion generated by the three-dimensional flow are adopted to mod- ify the sediment transport capacity formula originally devel- oped for estimating general scour. Dou’s study also includes some investigations on sediment incipient movement in local scour and includes some laboratory experiments. Roulund (2000) presents a comprehensive description on the flow around a circular pier and the development of the scour hole by numerical and experimental study. The numeri- cal model solves the three-dimensional RANS equations with use of the k −
(SST) turbulence closure model. The method is based on a full three-dimensional bed load formulation, including the effect of gravity. Based on the bed load calcula- tion, the change in bed level with time is calculated from the equation of continuity for the sediments. For the experiments, the scour development from a flat bed to the equilibrium of the scour hole was videotaped and this visual record was used to

observe the scour development. The evolution of the three- dimensional scour hole with time was obtained by combining the scour profiles from all angles around the pier. 6.3 NUMERICAL METHOD USED IN THIS STUDY In the present study, the three-dimensional flow chimera RANS Method of Chen et al. (1993, 1995a, 1995b, 1997) was used. First, the computational domain was divided into a num- ber of smaller grid blocks to allow complex configurations and flow conditions to be modeled efficiently through the judicious selection of different block topology, flow solvers, and bound- ary conditions. The chimera domain decomposition technique was used to connect the overlapped grids together by inter- polating information across the block boundaries. The Reynolds stresses were evaluated using the two-layer turbu- lence model of Chen and Patel (1988). The mean flow and turbulence quantities were calculated using the finite-analytical method of Chen, Patel, and Ju (1990). The SIMPLER/PISO pressure-velocity coupling approach of Chen and Patel (1989) and Chen and Korpus (1993) was used to solve for the pres- sure field. A detailed description of the multiblock and chimera RANS methods is given in Chen and Korpus (1993) and Chen, Chen, and Davis (1997). A useful summary of that method can be found in Nurtjahyo (2002). This summary discusses the governing equations, turbulence modeling (RANS or RANS equations), the boundary conditions on the pier surface, the river bottom, the outer boundaries, and the free water surface. The computer code has the ability to simulate the develop- ment of the scour hole around the pier as a function of time. This is done by including an erosion function linking the ver- tical erosion rate to the shear stress at the interface between the water and the soil. The program then steps into time by adjust- ing the mesh in the vertical direction after each time step as the scour hole develops. This option is not necessary to obtain the maximum shear stress before scour starts, since in this case the bottom of the river is kept flat. A typical run consists of the following steps: 1. Obtain the information for the problem: water depth, mean depth velocity at the inlet, pier size, and pier shape. 2. Calculate the Reynolds Number and Froude Number because they influence the size and distribution of the grid elements. 3. Generate the grid using a program called GRIDGEN (about 4 days’ worth of work). 4. The input consists of the Reynolds Number, the Froude Number, and the boundary conditions on all surfaces. The initial condition consists of the velocity profile at the inlet and is automatically generated by the program on the basis of the inlet mean depth velocity and the geometry. 5. Typical runs last 5 hours of CPU time on the Texas A&M University SGI supercomputer when only the 46 bed shear stress is required. The CPU time increases to 20 hours when the scour hole development needs to be simulated. 6. The output consists of the following parameters in three dimensions: velocity vectors, pressure, bed shear stress, and turbulent kinetic energy. 6.4 VERIFICATION OF THE NUMERICAL METHOD This verification was achieved by comparing the shear stresses predicted by the numerical method to those measured experimentally. The measurements were performed by Hjorth (1975) who investigated the distribution of shear stresses around a circular pier. The experiment was conducted in a rigid boundary flume. Two circular piers were used (diameter 0.05 m and 0.075 m). The study focused on two different velocities (0.15 m/s and 0.20 m/s) and two different depths of approach flow (0.1 m and 0.2 m). The shear stress was mea- sured by using a stationary hot film probe through the bottom of the flume flush with the bottom, then moving the position of the cylinder around the fixed probe. In that fashion, Hjorth obtained the shear stress at 35 different locations around the cylinder and created isostress lines by interpolation between the measurements. The numerical method was used to simu- late the experiments performed by Hjorth. The output was the distribution of the shear stresses on the flume bottom around the cylinder. The result of the experiment and the numerical simulation for the 0.075-m-diameter cylinder are compared on Figures 6.1 and 6.2. The very favorable comparison gave confidence in the validity of the numerical results. 6.5 SHALLOW WATER EFFECT: NUMERICAL SIMULATION RESULTS The objective of this parametric study was to obtain the relationship between the maximum bed shear stress τmax and water depth (Figure 6.3). One of the flume experiments was chosen to perform the simulation. The cylindrical pier with a diameter equal to 0.273 m was placed vertically in a 1.5-m-wide flume. The velocity was constant at 0.3 m/s and four different water depths were simulated: H = 0.546 m (or H/B = 2), H = 0.258 m (or H/B = 0.95), H = 0.137 m (or H/B = 0.5), and H = 0.060 m (or H/B = 0.22). The value of the Reynolds Number based on the diameter was Re = 81900 and the Froude Number based on the diameter was Fr = 0.1833. In order to reduce the amount of CPU time, one-half of the symmetric domain was chosen. The grid was divided into four blocks as shown in Figure 6.4. The grid was very fine near the pier and riverbed in order to apply the two-layer approach of the turbulence model. A few grid layers were placed within the viscous sublayer. The first step was to verify that the inlet velocity profile for the numerical simulation matched the experiment. The result was

47 shown in Figure 6.5. In the experiment, the thickness of the boundary layer was about 0.06 m for all cases. This observa- tion was also found in Gudavalli’s experiments (1997). The velocity vector around the pier is shown in Figure 6.6 for a shallow water case and a deep water case. The difference in velocity field can be observed on the figure, especially near the base of the pier where the horseshoe vortex is much stronger in the shallow water case. Figures 6.7 and 6.8 show the distribution of shear stress around the pier for different relative water depths of H/B = 0.2 and H/B = 2. As can be seen in the figures, the shear stress is higher in the case of the shallow water depth. This is explained as follows. When the water is deep, the velocity profile has the conventional shape shown in Figure 6.5. When the water depth becomes shallow, and if the mean depth velocity is kept constant, the velocity profile must curve faster towards the bottom of the profile because of the lack in verti- cal distance forced by the shallow water condition. This leads to a higher gradient of velocity near the bottom and, therefore, to a higher shear stress τ since τ is proportional to the veloc- ity gradient. Johnson and Jones (1992) made similar observa- tions on the influence of the pier diameter on the bed shear stress; these observations were based on experiments. Figure 6.9 shows the pressure field induced by the pier. If the pressure field is sufficiently strong, it causes a three- dimensional separation of the boundary layer, which in turn rolls up ahead of the pier to form the horseshoe vortex sys- tem. A blunt-nosed pier, for example, is a pier for which the induced pressure field is sufficiently strong to form the horse- shoe vortex system. 6.6 SHALLOW WATER EFFECT ON MAXIMUM SHEAR STRESS The maximum shear stress τmax is the maximum shear stress that exists on the riverbed just before the scour hole starts to develop. One way to present the data is to plot τmax/τmax(deep) as a function of H/B (Figure 6.10). The parameter τmax(deep) is the value of τmax for the deep water case and is given by Equa- tion 6.1. The shallow water correction factor, kw, is the ratio τmax/τmax(deep). The data points on Figure 6.10 correspond to the results of the four numerical simulations. (b) Numerical τ= (a) Experiment (Hjorth, 1975) -1 -0.5 0 X/B 0.5 Flow τ = 3 5 7 119 3 1 Figure 6.1. Comparison of bed shear stress (N/m2) distribution around a circular pier as calculated from an experiment by Hjorth (1975) and numerical computations (B = 0.075 m, V = 0.15 m/s, H = 0.1 m).

48 6.7 PIER SPACING EFFECT: NUMERICAL SIMULATION RESULTS The objective of this parametric study is to obtain the rela- tionship between the maximum bed shear stress τmax and pier spacing (Figure 6.11). One of the flume experiments was chosen to perform the numerical simulation. The cylindrical pier had a diameter of 0.16 m and was placed vertically in a 1.5-m-wide flume. The mean depth approach velocity was 0.33 m/s and the water depth is 0.375 m. Four different pier spacings were simu- lated: S/B = 6 (in the case of one pile in the flume), S/B = 3.12 (in the case of two piles), S/B = 2.34 (in the case of three piles), and S/B = 1.88 (in the case of four piles). The Reynolds Number based on diameter was Re = 52800 and the Froude Number based on diameter was Fr = 0.2634. The velocity between the piles became higher due to the de- creased spacing and the corresponding shear stress increases. (b) Numerical (a) Experiment (Hjorth,1975) τ = τ = 3 -1 -0.5 0 X/B Flow 0.5 5 7 9 11 3 1 H B Flow Figure 6.2. Comparison of bed shear stress distribution (N/m2) around a circular pier as calculated from an experiment by Hjorth (1975) and numerical computations (B = 0.075 m, V = 0.30 m/s, H = 0.2 m). Figure 6.3. Problem definition for water depth effect. By regression, the equation proposed for the correction factor kw giving the influence of the water depth on the max- imum shear stress is k ew H B= ( ) = + −τ τ max max ( . )deep 1 16 6 2 4

Figures 6.12 and 6.13 show the shear stress distribution for S/B = 1.88 and S/B = 6. 6.8 PIER SPACING EFFECT ON MAXIMUM SHEAR STRESS The maximum shear stress τmax is the maximum shear stress that exists on the riverbed just before the scour hole starts to develop. One way to present the data is to plot τmax/τmax(single) as a function of S/B (Figure 6.10). The param- eter τmax(single) is the value of τmax for the case of a single pier in deep water and is given by Equation 6.1. The pier spac- ing correction factor, ksp, is the ratio τmax/τmax(single). The data points on Figure 6.14 correspond to the results of the four numerical simulations. 49 By regression, the equation proposed for the correction factor ksp giving the influence of the pier spacing on the max- imum shear stress is 6.9 PIER SHAPE EFFECT: NUMERICAL SIMULATION RESULTS The objective of this parametric study was to obtain the relationship between the maximum bed shear stress τmax and the shape of rectangular piers (Figure 6.15). One of the flume experiments was chosen to perform the numerical k esp S B = ( ) = + −τ τ max max . . ( . ) single 1 5 6 3 1 1 Block # 4 Block # 2Block # 1 Block # 3 Figure 6.4. Grid system for the numerical simulation. 0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 0 .0 5 0 .1 0 .1 5 0 .2 0 .2 5 0 .3 0 .3 5 V e lo c ity (m /s ) D ep th (m ) F A N S E x p e r im e n t y /D = 2 E x p e r im e n t y /D = 1 E x p e r im e n t y /D = 0 .5 Figure 6.5. Velocity profile comparison between experiment and numerical simulation at the inlet.

50 simulation. A rectangular pier with a width of 0.061 m was placed vertically in the 1.5-m-wide flume. The velocity was constant and equal to 0.33 m/s and the water depth was 0.375 m. Four different pier aspect ratios were simulated: L/B = 1, 4, 8, and 12. The value of the Reynolds Number based on the width of the rectangular pier was Re = 20130 and the Froude Number based on the width of the rectan- gular pier was Fr = 0.4267. Examples of velocity fields are presented in Figures 6.16 and 6.17 for rectangular piers with aspect ratios equal to 0.25 and 4. Figures 6.18 to 6.20 show the maximum bed shear stress contours around rectangular piers with different aspect ratios: L/B = 0.25, 1, and 4. The location of the maximum bed shear stress was at the front corner of the rectangle. It was found that the maximum bed shear stress, τmax, was nearly constant for any aspect ratio above one. The value of τmax X Y Z (a). H/B=0.2 X Y Z (b). H/B=2 Figure 6.6. Velocity vector around pier for (a) H/B = 0.2 and (b) H/B = 2. 4.97 7.33 0.25 X/B-1 -0.5 0 0.5 1 Flow 0.56 1.03 2.21 0.78 0.25 0.35 0.45 0.55 0.65 0.75 1.05 0.85 0.25 X/B-1 -0.5 0 0.5 1 Flow τ = 0.16 0.23 0.09 0.36 -0.37 -0.37 -0.30 -0.43 -0.43 X/D Y/ D -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Pier Flow Figure 6.8. Initial bed shear stress distribution (N/m2) around the pier (H/B = 2, V = 0.3m/s). Figure 6.9. Normalized pressure (p/ρu2) contours for H/B = 2 on the riverbed. Figure 6.7. Initial bed shear stress distribution (N/m2) around the pier (H/B = 0.2, V = 0.3m/s).

51 increased, however, when L/B became less than one. Figures 6.16 and 6.17 indicate that the flow pattern around the rec- tangular pier for L/B = 0.25 is quite different from the pattern for L/B = 4 where the flow is separating at the sharp corner. For L/B = 0.25 the flow is allowed to go behind the pier while the flow for L/B = 4 follows the side of the pier. In the case of L/B = 4, the length of the flow separation is about 1 B from the corner; this may explain that τmax is independent of the pier length for L/B > 1. On the contrary, for L/B < 1, there is no region of separated flow and the decreasing pressure behind the pile may increase the velocity around the corner. 6.10 PIER SHAPE EFFECT ON MAXIMUM SHEAR STRESS The maximum shear stress τmax is the maximum shear stress that exists on the riverbed just before the scour hole starts to develop. One way to present the data is to plot ksh = τmax/τmax(circle) as a function of L/B (Figure 6.21). The param- eter τmax(circle) is the value of τmax for the case of a circular pier in deep water and is given by Equation 6.1. The pier spacing correction factor ksh is the ratio τmax/τmax(circle). The data points 0 1 2 3 4 5 6 7 8 9 0 0.5 1 1.5 2 2.5 H/B k w H B Figure 6.10. Relationship between kw (= τmax /τmax(deep)) and H/B. Flow B S Figure 6.11. Problem definition of pier spacing effect. 0.30 0.47 0.64 0.81 0.98 1.14 1.82 0.98 0.81 0.64 0.47 0.301.31 1.48 X/B Y/ B -0.5 0 0.50 0.2 0.4 0.6 0.8 1 Flow τ = Figure 6.12. Initial bed shear stress distribution (N/m2) around the pier (S/B = 1.88, H = 0.38m, V = 0.33m/s).

on Figure 6.21 correspond to the results of the seven numer- ical simulations. The correction factor for shape effect ksh is given by the following equation, which was obtained by regression of the data points on Figure 6.21: k esh L B = + − 1 15 7 6 4 4 . ( . ) 52 6.11 ATTACK ANGLE EFFECT: NUMERICAL SIMULATION RESULTS The objective of this parametric study was to obtain the relationship between the maximum bed shear stress τmax and the angle of attack α defined as the angle between the flow direction and the pier direction (Figure 6.22). One of the flume experiments was chosen to perform the numerical sim- ulation. A rectangular pier with a width of 0.061 m was placed vertically in a 1.5-m-wide flume. The velocity was 0.33 m/s, the aspect ratio of the pier was L/B = 4, and the water depth was 0.375 m. Four different attack angles were investigated: α = 15, 30, 45, and 90 degrees. Based on this data, the value of the Reynolds Number based on the width of the rectangular pier was Re = 20130 and the Froude Num- ber based on the width of the rectangular pier was Fr = 0.4267. The bed shear stress distributions for attack angles of 15, 30, and 45 degrees are shown in Figures 6.23 to 6.25. These figures indicate that the value of the maximum bed shear Flow L B Figure 6.15. Problem definition for the shape effect. 0.30 0.40 0.49 0.59 0.69 0.78 0.88 1.17 X/B Y/ B -0.5 0 0.50 0.2 0.4 0.6 0.8 1 Flow τ = Figure 6.13. Initial bed shear stress distribution (N/m2) around the pier (S/B = 6, H = 0.38m, V = 0.33m/s). 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0 1 2 3 4 5 6 7 S/B k sp B S Flow Figure 6.14. Relationship between ksp (= τmax/τmax(single)) and S/B for deep water H/B > 2).

53 X/B Y/ B -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 X/B Y/ B -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Figure 6.16. Velocity field around a rectangular pier with L/B = 0.25. Figure 6.17. Velocity field around a rectangular pier with L/B = 4. 0.33 0.33 0.67 1.00 1.34 1.68 2.35 0.33 0.33 0.33 0.33 X/B Y/ B -1 -0.5 0 0.5 1 1.50 0.5 1 1.5 2 5.10 Flow = Figure 6.18. Bed shear stress contours (N/m2) around a rectangular pier (L/B = 0.25). 0.33 0.33 0.47 0.74 0.88 1.15 1.34 1.56 0.33 0.33 0.33 0.33 X/B Y/ B -1 -0.5 0 0.5 1 1.50 0.5 1 1.5 2 Flow Flow = Figure 6.19. Bed shear stress contours (N/m2) around a rectangular pier (L/B = 1). stress tends to increase with the attack angle. They also show that the location of the maximum shear stress moves back- ward along the side of the pier as the attack angle increases. 6.12 ATTACK ANGLE EFFECT ON MAXIMUM SHEAR STRESS The maximum shear stress τmax is the maximum shear stress that exists on the riverbed just before the scour hole starts to develop. One way to present the data is to plot τmax/τmax(0 degree) as a function of α (Figure 6.26). The param- eter τmax(0 degree) is the value of τmax for the case of a pier in line with the flow in deep water and is given by Equation 6.1. The attack angle correction factor, ksh, is the ratio τmax/τmax(0 degree). The data points on Figure 6.21 correspond to the results of the five numerical simulations. By regression, the equation 0.33 0.42 0.51 0.59 0.68 0.77 0.95 1.12 1.30 1.56 0.33 0.33 X/B Y/ B -1 -0.5 0 0.5 1 1.50 0.5 1 1.5 2 Flow = Figure 6.20. Bed shear stress contours (N/m2) around a rectangular pier (L/B = 4).

proposed for the correction factor ka giving the influence of the attack angle on the maximum bed shear stress is ka = ( ) = +  ττ αmax max . deg . ( . ) 0 1 1 5 90 6 5 0 57 54 6.13 MAXIMUM SHEAR STRESS EQUATION FOR COMPLEX PIER SCOUR In the previous sections, individual effects on the maximum shear stress are studied by numerical simulations. A series of figures and equations are given to quantify the corresponding correction factors. However, bridge piers are likely to exhibit a combination of these effects and recommendations are needed to combine these effects in the calculations. It is rec- ommended that the correction factors be multiplied in order to represent the combined effect. This common approach implies that the effects are independent and has been used in many instances. The proposed equation for calculating the maximum shear stress for a complex pier before the scour process starts is τ ρmax . log Re ( . )= × −    k k k k Vw sp sh a 0 094 1 1 10 6 6 2 1.361 .05 1.55 1.88 2.11 1 .55 1 .36 1.05 2.26 1.8 8 X/B Y/ B -2 -1 0 1 2 3 4 5 6-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Pier Flow = Figure 6.23. Bed shear stress (N/m2) contours for an attack angle equal to 15 degrees. 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 L /B k sh F l o w L B B L sh ek 4 715.1 − += Figure 6.21. Relationship between ksh (= τmax/τmax(circle)) and L/B for deep water (H/B > 2). Flow L B α Figure 6.22. Definition of the angle of attack problem. 0.93 1.11 1 .47 0.93 1.29 1.65 2.00 0.24 0.75 0.75 0.93 1.2 9 1.83 2.36 0.24 0.93 2 .90 X/B Y/ B -2 -1 0 1 2 3 4 5 6-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 = Flow Figure 6.24. Bed shear stress (N/m2) contours for an attack angle equal to 30 degrees.

55 where • B is the pier width (m); • H is the water depth (m); • V is the upstream velocity (m/s); • ρ is the density of water (kg/m3); • α is the attack angle (in degrees); • Re is the Reynolds Number, defined as • v is the kinematic viscosity (m2/s); • kw is the correction factor for the effect of water depth, defined as and the equation is kw = 1 + • ksp is the correction factor for the effect of pier spacing, defined as and the equation is ksp = 1 + • ksh is the correction factor for the effect of pier shape, defined as and the equation is ksh = 1.15 + • ksh = 1 for circular shape; and • ka is the correction factor for the effect of attack angle, 7 4 e L B − ; k circlesh = ( ) τ τ max max 5 1 1 e S B − . ; ksp = ( ) τ τ max max single 16 4 e H B − ; k deepw = ( ) τ τ max max Re = VB v ; 3.27 3.002 .52 2.32 2.03 1.611.331 . 08 0.33 1 .08 1. 33 1.61 2.5 2 2.03 0. 33 1.33 0.33 0.33 2 .52 X/B Y/ B -2 -1 0 1 2 3 4 5 6-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Pier = Flow Figure 6.25. Bed shear stress (N/m2) contours for an attack angle equal to 45 degrees. 0 0 . 5 1 1 . 5 2 2 . 5 3 0 0.2 0.4 0.6 0.8 1 1.2 α/90 k α F l o w L B α 5 7.0 9 0 5.11 += α ak Figure 6.26. Relationship between ka (= τmax/τmax(0 degree)) and α for deep water (H/B > 2). defined as and the equation is ka = 1 + 1 5 90 0 57 . .α  ka = ( ) τ τ max max deg0