BAGNOLD FORMULA REVISITED: INCORPORATING PRESSURE GRADIENT INTO ENERGETICS MODELS

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1 BAGNOLD FORMULA REVISITED: INCORPORATING PRESSURE GRADIENT INTO ENERGETICS MODELS Qun Zhao and James T. Kirby 2, M. ASCE Abstract: In this paper, e examine a reformulation of the Bagnold sediment transport formula, based on using friction velocity to express bottom shear stress. This modification allos the transport formulation to retain the effect of phase lag beteen free stream velocity and bottom stress, neglected in Bagnold s original formula and usually associated ith flo acceleration (or, in other ords, pressure gradient). Friction velocity is computed using a linearized -D bottom boundary layer model. The modified Bagnold model can predict net onshore sediment transport for asymmetric, zero skeness aves. Results for analytical aveforms are compared ith previous ork based on discrete particle simulations and to-phase flo methods and sho qualitative consistency. INTRODUCTION The Bagnold Model Bagnold (963, 966) derived a stream-based sediment transport model. In that model, Bagnold assumes the sediment is transported in to modes, i.e., the bedload transport and the suspended transport. The bedload sediment is transported by the flo via grain to grain interactions, the suspended sediment transport is supported by fluid flo through turbulent diffusion. The total load sediment transport rate i reads (Bagnold, 966) i = ɛ b tan φ tan β + ɛ s( ɛ b ) (/u b ) tan β ω () here ω is the available fluid poer, is the fall velocity of sediment. ɛ b and ɛ s are the bedload and suspended load efficiencies, respectively. They both smaller than one. tan β is the bottom slope, and φ is the particle friction angle. The available fluid poer ω is the ork done by the bottom shear stress τ b ω = τ b u b (2) here u b is the near bed free stream velocity. The bottom shear stress is parameterized using the quadratic drag la τ b = ρc f u b u b (3) ith c f the bottom friction coefficient. Tianjin Research Institute for Water Transport Engineering, Ministry of Transportation of China, 37 The Second Road of Xingang, Tanggu, Tianjin 3456, P. R.China. zhao@coastal.udel.edu 2 Center for Applied Coastal Research, University of Delaare, Neark, DE 976, U.S.A. kirby@coastal.udel.edu Zhao and Kirby

2 Wave-averaged Bagnold Model-The BBB Concept Substituting (2)-(3) into (), and considering the bottom slope effect, Bailard and Inman (98) obtained the total sediment transport rate i ritten in terms of free stream velocity i = ρc f ɛ b tan φ + ρc f ɛ s ( ɛ b ) u b 2 u b tan β tan φ u b 3 u b 3 u b ɛ s( ɛ b ) tan β u b 5 The first bracket in (4) represents the bedload sediment transport, and the second bracket represents the suspended sediment transport. Both contain the sediment transport in the direction of the instantaneous velocity(the first and third term), and the sediment transport don slope (the second and fourth term). In (4), the near bed free stream velocity u b is taken to be the near bed orbital velocity. Bailard (98) took a time-average of (4) for the ave period T to obtain the total net sediment transport rate due to aves and currents. ɛ b = ρc f tan β tan φ tan φ ɛ s ( ɛ b ) + ρc f ɛ s( ɛ b ) tan β (5) Here < f > denotes the ave-averaging of arbitrary variable f, < f >= T T (4) fdt (6) Equation (5) is often referred to as the BBB energetics model, ith BBB representing Bagnold (963,966), Boen(98) and Bailard (98). Limitation of the Bagnold-type Models The Bagnold model ()-(3) and its ave-averaged version (5) relate the total sediment transport rate to the near bed free stream velocity. This model lacks information on flo acceleration or pressure gradient effects ithin the bottom boundary layer, and, in particular, predicts zero sediment transport in ave fields ith zero velocity skeness. Recent field observations (Gallagher et al., 998; Elgar et al., 2) have shon net onshore sediment transport beteen storms (manifested mainly by net onshore motion of shore-parallel bars) hen unbroken aves are strongly skeed but underto over the bar crest is eak. The BBB transport formula successfully predicts offshore sediment transport and bar crest movement during energetic ave events, buts fail to predict the onshore bar motion in loer energy conditions hen calibrated ith coefficients that are appropriate for offshore transport (Gallagher et al., 998). Based on field observations, Elgar et al. (2) noticed that the onshore sediment transport (or bar crest motion) is strongly correlated to cross shore gradient in a ave-averaged acceleration skeness, indicating that the local transport rate is likely to be directly dependent on this statistic as ell. Numerical computations based on a discrete particle method (Drake and Calantoni, 2) have also shon that net transport can occur under zero-skeness aves, and is correlated ith flo 2 Zhao and Kirby

3 acceleration. Base on this as a starting point, Drake and Calantoni (2) introduced an ad hoc extra term a spike =< ( u b / t) 3 > / < ( u b / t) 2 > into the BBB model to account for the difference in the magnitude of the acceleration under the front and back of the ave. The extra sediment transport rate to be added into the BBB formula (5) due to acceleration as given as { Ka (a = spike sgna spike a crit ) a spike >= a crit (7) a spike < a crit here a crit is the critical value of a spike that must be exceeded before acceleration enhances the sediment transport, and K a is the model parameter. The total net sediment transport rate is then ɛ b = ρc f tan φ + ρc f ɛ s ( ɛ b ) tan φ tan β ɛ s( ɛ b ) tan β + (8) The modified BBB formula (8) has been tested using measured near bed velocities (Hoefel and Elgar, 23) and shoed favorable results to the original BBB formula (5) hen comparing ith data. Long and Kirby (23) have also used an extended Bagnold model, incorporating an instantaneous acceleration term, to model transport in a phase-resolving Boussinesq model calculation, and ere able to predict qualitatively accurate onshore bar migration in a model prediction using only incident ave conditions measured offshore. A REFORMULATED BAGNOLD MODEL Despite the success of the extended Bagnold models discussed above, it is clear that the added acceleration (or pressure gradient) effects are incorporated in an ad hoc manner hich does not have a clear theoretical foundation. More recently, attention has turned to a more direct examination of the bottom boundary layer mechanics and the relation beteen unsteady free-stream velocity and resulting bed shear stress. Hsu and Hanes (24) have considered the calculation of bed stress using artificial skeed and asymmetric free stream velocities, and have shon that predictions of a detailed to-phase transport model can be recovered by a simple application of the Meyer-Peter Muller formula using the calculated bed stress. Henderson et al. (24) have used a similar approach and have shon, using measured data to drive a -D vertical boundary layer model, that both erosional and accretionary behavior of sand bar crests can be modelled and that good agreement ith field observation can be obtained. Long et al (24) have used a similar approach but employ free stream data predicted by a Boussinesq model to similarly predict onshore bar migration using computed bed stress and the Meyer-Peter Muller transport formula. The Bagnold model and its ave-averaged version, the BBB model, do not account for flo ith strong acceleration or pressure gradient. This limitation is introduced hen the bottom shear stress is parameterized using the near bed free stream velocity through the quadratic drag la (3) through Reynolds similarity. We recognize that for flos ith a free surface, here gravitational forces (pressure gradient due to free surface) must be considered, the parameterization of the bottom shear stress using (3) is not valid (Schlichting, 96). This is also pointed out by Nielsen (22) and Nielsen and Callaghan (23). They 3 Zhao and Kirby

4 acknoledge that the Bagnold type formula ritten in terms of the odd moments of near bed free stream velocity fail spectacularly under some real aves. Based on laminar bottom boundary layer theory, Nielsen (992) obtains the bottom shear stress ritten in terms of near bed free stream velocity and a phase shift ϕ τ, in hich ϕ τ is the phase lag beteen the bottom shear stress and the free stream velocity, and found to be around 4 degree based on the data they tested. Here, e employ a more brute force approach and compute the bottom shear stress and friction velocity u using a bottom boundary layer (BBL) model (see Appendix), in the spirit of Hsu and Hanes (24). In the folloing, e replace (3) ith the expression τ b = ρ u u (9) hich is the fundamental definition of friction velocity. Hoever, the parameterization for the fluid poer ω is not straightforard and needs further discussion. Here, e follo Bagnold (963) and consider that the ork done on the sediments is at the center of pressure, and the velocity at that point is parameterized as u = c r u. c r needs to be determined experimentally. Then the ork done by the bottom shear stress is ω = τ b u = ρc r u 2 u () Therefore, the modified Bagnold formula is i = ρ ɛ bc r tan φ + ρ ɛ s( ɛ b )c r u 2 u tan β tan φ u 3 u 3 u ɛ s( ɛ b ) tan β u 5 () It is seen that () is very similar to (4), except e replace the free stream velocity u b by the friction velocity u. It is orthhile to point out that the effect of pressure gradient is implicitly included in the friction velocity u, but not in the free stream velocity u b. By performing ave-averaging, e obtain the modified BBB model = ρ ɛ bc r tan β tan φ tan φ + ρ ɛ s( ɛ b )c r ɛ s( ɛ b ) tan β (2) It ill be shon in the next section that using results in net onshore sediment transport for asymmetric, zero skeness aves, hereas using produces no net sediment transport. Finally, the ave-averaged net bedload flux < q > is related to the immersed eight transport rate (Drake and Calantoni, 2), < q >= g(ρ s ρ) ρ s (3) 4 Zhao and Kirby

5 RESULTS In this section e examine the performance of the BBB type formula ritten in terms of the near bed free stream velocity and the friction velocity, respectively. We study the instantaneous and ave-averaged bedload sediment transport under zero-skeness sa-tooth shape aves. Sa-tooth shape aves are of interest because they are similar to broken aves, and the BBB formula fails to predict any onshore sediment transport due to zero skeness. Hoever, both field observations and numerical modelings using discrete particle method (Drake and Calantoni, 2) and to-phase flo approach (Hsu and Hanes, 24) sho extensive sediment transport due to flo acceleration (or pressure gradient). In Drake and Calantoni (2), the near bed free stream velocity u b is driven by the pressure gradient F (t), and F (t) follos F (t) 4 n= 2 sin (n + ) 2π n T t + nφ and φ is defined as the aveform parameter. The typical sa-tooth ave is that of φ = π 2. In Hsu and Hanes (24), the sa-tooth shaped free stream velocitiy is described by u b (t) = U s 5 n= = U s 4 n= sin 2n 2 n sin n 2π t + (n )π T (n + ) 2π T t + nπ in their Equation (7), ith U s the velocity amplitude. Hoever, e found that the near bed free stream velocity in Drake and Calantoni (2) do not response to the pressure gradient force F (t) linearly, and the ave form discussed in Hsu and Hanes (24) do not correspond to equation (5)(Equation (7) in their paper). In the present ork, the sa-tooth ave is constructed using 4 u b (t) = 2 sin (n + ) 2π n T t (6) n= This result roughly corresponds to the sa-tooth ave ith aveform parameter φ = π 2 in Drake and Calantoni (2). Figure shos the results for this sa-tooth ave. The upper panel of Figure shos the free stream velocity (solid line) and the flo acceleration (dashed line). The loer panel of Figure shos corresponding friction velocity u 2 u (solid line) and near bed velocity u b 2 u b (dashed line). It shos in Figure that u b 2 u b is symmetric, and equals to zero upon ave-averaging. Whereas u 2 u is asymmetric and the ave-average is positive (onshore directed). If e consider the bed slope is very small, and bed-load sediment transport dominates, then u b 2 u b and u 2 u are proportional to the time-dependent bed load flux q. And if e compare this result ith that shon in Drake and Calantoni (2), e see that the bed-load flux computed using the friction velocity u and (2) is very similar to the results of Drake and Calantoni (2). Moreover, this result is also qualitatively similar to that presented by Hsu and Hanes (24) in their Figure 2, if e integrate their results vertically. Hoever, it is noticed that Figure shos virtually no sediment transport here the (4) (5) 5 Zhao and Kirby

6 PSfrag replacements u b (m/s) u b t (m/s2 ) m 3 /s t/t Fig.. Results corresponding to Drake and Calantoni (2), φ = π/2. Upper panel: Free stream velocity (solid line) and flo acceleration (dashed line). Loer panel: u 2 u (solid line) shoing net on shore sediment transport hen averaged over ave period T, and u b 2 u b (dashed line) shoing no net sediment transport due to zero skeness. The magnitudes of the variables are adjusted to fit the figure. free stream velocity is zero, hereas Hsu and Hanes (24) shoed limited rate of sediment transport. We think this may be due to the fact that in the present ork e neglected the nonlinear convective terms in the BBL equations, so that e may miss some phase information in the bottom shear stress. Nonetheless, the important feature is that the modified Bagnold formula can predict net onshore sediment transport under asymmetric zero skeness aveforms provided the bottom friction is parameterized using the friction velocity. For comparison, a case of a symmetric, skeed ave form is shon in Figure 2. The near bed free stram velocity is calculated using u b (t) = 4 n= 2 sin (n + ) 2π n T t + (n π) and it corresponds to the aveform parameter φ =, Figure 5 in Drake and Calantoni (2) and Figure 8-9 in Hsu and Hanes (24). For this case, both u 2 u and u b 2 u b predict net onshore sediment transport due to ave skeness. Next e compare the ave-averaged bedload flux < q > versus the third moments of free stream velocity for different ave forms on a flat bottom. The results are computed using (2) and (3). The ave period is T = 6.s and the maximum free stream velocities are.5,.75,.,.25,.5m/s. The ave forms roughly correspond to φ =, π, π in Drake 4 2 and Calantoni (2), though a precise match of the ave forms is found difficult to achieve. Parameters generally suggested for the use of the BBB formula (Gallagher et al., 998) (7) 6 Zhao and Kirby

7 PSfrag replacements u b (m/s) u b t (m/s2 ) m 3 /s Fig. 2. Results corresponding to Drake and Calantoni (2), φ =. Both cases sho net onshore sediment transport due to ave skeness. Legends are the same as the previous figure. t/t are used here ith tan φ =.63 and ɛ b =.2. The parameter c r in () is chosen to be 5 for the present study. We see that Figure (3) shos strikingly similar pattern to that shon in Drake and Calantoni (2) for their Figure (7). Hoever, our results are significantly smaller than those presented by Drake and Calantoni (2). The reason is not clear, but e notice that the Drake and Calantoni (2) results ere obtained using a unusually large bedload efficiency, ɛ b =.3, hereas physics requests the bedload efficiency to be less than one. Besides, field comparisons using the Drake and Calantoni (2) formula (4)-(8) shoed that their K a is orders larger than the calibrated value(see Hoefel and Elgar, 23, Long and Kirby 23). CONCLUSIONS In this paper e rederive the Bagnold formula and e found that the net onshore directed sediment transport is correlated to the bottom shear stress. The BBB-type formula ritten in terms of odd moments of free stream velocity as derived using the parameterized bottom shear stress for quasi-steady flo. Because this parameterization for bottom shear stress is not valid for flo ith acceleration, the BBB-type formula as unable to predict net onshore sediment transport under zero skeed satooth aves. Using bottom shear stress computed from the BBL equations, the corrected BBB-type formula can predict net onshore sediment transport. Results for analytical aveforms are discussed and compared ith previous orks using discrete particle (Drake and Calantoni, 2) and to-phase flo method (Hsu and Hanes, 24) and shoed qualitative consistency. 7 Zhao and Kirby

8 < q > (kg/m/s) PSfrag replacements. φ=.5 φ=π/4 φ=π/ (m3 /s 3 ) Fig. 3. Computed bedload flux versus odd moments of the free stream velocity for different ave forms. The ave period is T = 6.s and the maximum free stream velocities are.5,.75,.,.25,.5 m/s. APPENDIX: SOLVING THE BOTTOM BOUNDARY LAYER EQUATIONS The continuity and momentum equations inside the bottom boundary layer (BBL) read, and u x + z = (8) u t + u u x + u z = u b t + u u b b x + τ zx ρ z here u b and u are the near bed free stream velocity and velocity inside the BBL, respectively. The boundary conditions for (8) and (9) are ith z the bed level. (9) u = ; = ; at z = z (2) u = u b ; = ; at z (2) Assuming the nonlinear convective terms are higher order terms, e obtain the linearized BBL equation (Trobridge and Madsen, 984), u t = u b t + τ zx ρ z (22) The Boussinesq assumption relates the shear stress to the gradient of velocity through τ zx = ρ(ν + ν T ) u z (23) 8 Zhao and Kirby

9 here ν and ν T are the kinematic and turbulent eddy viscosity, respectively. And ν T is computed using a turbulence model. In general, turbulence models that record flo history, such as the k ɛ type models, are more accurate than those don t. Hoever, e also notice that virtually almost all turbulence models impose the log la (also termed as the all function ) at the all. And previous researches (Wilcox, 998 for example) sho that the BBL results are not sensitive to turbulence models used near the bottom. Therefore, in this research, e ill use the more efficient mixing length model. Then the eddy viscosity is computed through the friction velocity ν T = κ u z (24) here κ =.4 is the Karman coefficient, and z is the distance from bed. denotes the absolute value of a variable. Imposing the log la, the friction velocity relates to the near bed velocity ( ) z u = κu ln (25) z here u is the BBL velocity at the first numerical grid point folloing Launder and Spadling (974), and z is the bed level and normally taken as z = k N /3. The bed roughness k N relates to the grain diameter d through k N = Nd. Here d =.mm and k N = 5d is chosen for this study. The above linearized BBL problem is a typical diffusion problem. Without loose of generality, e assume the the boundary layer thickness δ = 5cm and solve it using the FTCS (forard time, center space) scheme. 6 grid points are solved in the vertical direction and the time step is computed so that the diffusion in one time step is smaller than one grid size. ACKNOWLEDGMENTS This research as funded by the National Oceanographic Partnership Program (N ) and the Office of Naval Research, Coastal Geosciences Program (N ). Qun Zhao acknoledges the financial support from Coastal and Estuarine Division, Tianjin Research Institute for Water Transport Engineering, Ministry of Transportation of China. REFERENCES Bagnold, R. A. (963): Mechanics of marine sedimentation. The Sea: Ideas and Observations, M. N. Hill ed, Interscience, Vol. 3, Bagnold, R. A. (966): An approach to the sediment transport problem from general physics. Prof. Paper 422-, U. S. Geol. Surv. Bailard, J. A. and D. L. Inman (98): An energetics bedload transport model for a plane sloping beach: local transport. J. Geophys. Res., 86, Bailard, J. A. (98): An energetics total load sediment transport model for a plane slopping beach. J. Geophys. Res., 86 (C), Boen, A. J. (98): Simple models of nearshore sedimentation: Beach profiles and longshore bars. In Coastline of Canada, -, Halifax. Geological Survey of Canada. 9 Zhao and Kirby

10 Drake, T. G. and J. Calantoni (2) Discrete particle model for sheet flo sediment transport in the nearshore. J. Geophys. Res., 6 (C9), Elgar, S., E. L. Gallagher and R. T. Guza (2): Nearshore sandbar migration. J. Geophys. Res., 6 (C6), Fredsoe J. and R. Deigaard (992): Mechanics of coastal sediment transport. World Scientific. 369pp. Gallagher, E. L. S. Elgar and R. T. Guza (998): Observation of sand bar evolution on a natural beach. J. Geophys. Res., 3 (C2), Henderson, S. J. S. Allen and P. A. Neberger (24): Nearshore sandbar migration predicted by an eddy-diffusive boundary layer model. J. Geophys. Res., 9 (C624). Hoefel, F. and S. Elgar (23): Wave-induced sediment transport and sandbar migration. Science, 299, Hsu, T. -J. and D. M. Hanes (24): The effect of ave shape on sheet flo sediment transport. J. Geophys. Res., 9, C525. Launder B. E. and D. B. Spalding (974): The numerical computation of turbulent flos. Comp. Meth. Appl. Mech. Engineering, 3, pp Long, W. and J. T. Kirby (23): Cross-shore sediment transport model based on the Boussinesq equations and an improved Bagnold formula, Proceedings Coastal Sediments 3, ASCE, Clearater Beach. Long, W., T.-J. Hsu and J. T. Kirby (24): Modeling cross-shore sediment transport processes ith a time domain Boussinesq model, Proc. 29th Int. Conf. Coastal Engrng., Lisbon, September, in press. Nielsen, P. (992): Coastal bottom boundary layers and sediment transport. World Scientific, Singapore. 324 pp. Nielsen, P. (22): Shear stress and sediment transport calculations for sash zone modelling. Coastal Engineering, 45, Nielsen, P. and Callaghan, D. P. (23): Shear stress and sediment transport for sheet flo under aves. Coastal Engineering, 47, Schlichting, H. (96): Boundary layer theory. McGra-Hill Book Company. 647pp. Trobridge, J. and O. S. Madsen (984): Turbulent ave boundary layers. Model formulation and first-order solution. J. Geophys. Res., 89 (C5), Wilcox, D, C.(998): Turbulence modeling for CFD. DCW Industries. 54pp. Zhao and Kirby

ANALYTIC SOLUTION FOR THE FORCED MEAN CROSS-SHORE FLOW IN THE SURF ZONE

ANALYTIC SOLUTION FOR THE FORCED MEAN CROSS-SHORE FLOW IN THE SURF ZONE Thomas C. Lippmann 1, Assoc. MASCE Analytical solutions to the forced horizontal momentum equations are found for the local vertical