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Scour depth prediction at the base of longitudinal walls: a combined experimental, numerical, and field study

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A series of small- and large-scale experiments and numerical simulations were carried out to produce datasets for maximum scour depth at the base of longitudinal structures. The small-scale experimental datasets were employed to develop an empirical relationship for predicting maximum local scour depth. The measured datasets for scour patterns and depths were used to validate the accuracy of our morphodynamics model, the Virtual Flow Simulator (VFS-Geophysics). The large-scale experimental measurements and numerical simulations lead to the development of another empirical relationship to estimate the maximum general scour depth near the longitudinal structures in large-scale meandering streams/rivers. The premise of this study is that the maximum scour depths obtained from the two equations can be used independently to represent local and general scour at the base of longitudinal walls in meandering rivers. However, for a more conservative prediction, the two can be linearly summed up to obtain a total maximum scour depth. The presented correlations illustrate regression goodness of r2 = 0.63 and 0.82 for the maximum local and general scour depth equations, respectively. Nonetheless, the developed equations are valid within a specific range of parameter for the sediment material, flow field, and waterway characteristics.

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Acknowledgements

This work was supported by National Cooperative Highway Research Program Grants NCHRP-HR 24–33 and 24–36.

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Appendices

Appendix 1: Governing equations of the numerical model

The detailed description of VFS-Geophysics mathematical formulations, hydro-morphodynamic coupling method, and boundary conditions can be found in [24]—see also [16, 17, 19, 22]. Herein, we briefly outline the equations that govern the hydrodynamics and morphodynamics of the model.

1.1 Hydrodynamic model

The hydrodynamic model solves numerically the unsteady, three–dimensional, incompressible Reynolds-averaged Navier–Stokes and continuity equations. The governing equations in generalized curvilinear coordinates, read as follows [20]:

$$ J\frac{{\partial U^{j} }}{{\partial \xi^{j} }} = 0 $$
(12)
$$ \frac{1}{J}\frac{{\partial U^{i} }}{\partial t} + \frac{{\xi_{{x_{\varrho } }}^{i} }}{J}\left( {\frac{\partial }{{\partial \xi^{j} }}\left( {U^{j} u_{\varrho } } \right) + \frac{\partial }{{\partial \xi^{j} }}\left( {\frac{{\xi_{{x_{\varrho } }}^{j} p}}{J}} \right) - \frac{1}{Re}\frac{\partial }{{\partial \xi^{j} }}\left( {\frac{{g^{jk} }}{J}\frac{{\partial u_{\varrho } }}{{\partial \xi^{k} }}} \right) + \frac{{\partial \tau_{\varrho j} }}{{\partial \xi^{j} }}} \right) = 0 $$
(13)

where \( \left\{ {x_{{ ^{i} }} } \right\} \) and \( \left\{ {\xi^{i} } \right\} \) are the Cartesian and generalized coordinates (i = 1,2, and 3), respectively, J is the Jacobian of the geometric transformation \( J = \partial \left( {\xi^{1} \xi^{2} ,\xi^{3} } \right)/\partial \left( {x_{1} ,x_{2} ,x_{3} } \right) \), \( \xi_{{x_{\varrho } }}^{i} = \partial \xi^{i} /\partial x_{\varrho } \) are the metrics of the geometric transformation, \( \left\{ {u_{\varrho } } \right\} \) are the Cartesian velocity components, \( U^{i} = \xi_{{x_{m} }}^{i} u_{m} /J \) are the contravariant components of the volume flux vector, \( g^{jk} = \xi_{{x_{\varrho } }}^{j} \xi_{{x_{\varrho } }}^{k} \) are the components of the contravariant metric tensor, p is the pressure, \( \tau_{ij} \) is the Reynolds stress tensor, \( Re \) is the Reynolds number (\( = \frac{{hU_{m} }}{\nu } \)), h is the mean-flow depth, \( U_{m} \) is the mean flow velocity, and \( \nu \) is the kinematic viscosity of water. The above equations are closed using the \( k - \omega \) model [40] to calculate the eddy viscosity. The turbulence closure equations in generalized curvilinear coordinates can be found in [24].

The governing equations are discretized in space on a hybrid staggered/non-staggered grid arrangement [11] using the second-order accurate QUICK scheme for the convective terms along with second-order accurate, three-point central differencing for the divergence, pressure gradient and viscous-like terms. The time derivatives are discretized using second-order backward differencing [16]. The discrete mean flow equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton–Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation.

1.2 Morphodynamic model

The temporal variation of the river bed elevation is governed by the Exner-Polya equation [36]:

$$ \left( {1 - \gamma } \right)\frac{{\partial z_{b} }}{\partial t} + \nabla \cdot {\mathbf{q}}_{{ _{BL} }} = D_{b} - E_{b} $$
(14)

where \( \gamma \) is the sediment material porosity, \( z_{b} \) is the bed elevation, \( \nabla \) denotes the divergence operator, \( {\mathbf{q}}_{BL} \) is the bed-load flux vector, \( D_{b} \) is the net deposition onto the bed, and \( E_{b} \) is the net entrainment from the bed cell. The motion of sediment in suspended load is governed by the following convection–diffusion equation [41, 42]:

$$ \frac{1}{J}\frac{{\partial \left( {\rho \psi } \right)}}{\partial t} + \frac{\partial }{{\partial \xi^{j} }}\left( {\rho \psi \left( {U^{j} - W^{j} \delta_{i3} } \right)} \right) = \frac{\partial }{{\partial \xi^{j} }}\left( {\left( {\sigma_{L}^{*} \mu + \sigma_{T}^{*} \mu_{t} } \right)\frac{{G^{jk} }}{J}\frac{\partial \psi }{{\partial \xi^{k} }}} \right) $$
(15)

where \( W^{j} = \left( {\xi_{3}^{j} /J} \right)w_{s} \) is the contravariant volume flux of suspended sediment, \( \sigma_{L}^{*} \) and \( \sigma_{R}^{*} \) are the laminar (\( = 1/200 \)) and turbulent (= \( 4/3 \)) Schmidt numbers, \( \rho \) is the density of water, and \( w_{s} \) is the settling velocity of non-spherical sediment particles which is computed by van Rijn’s formula [39].

At the mobile sediment/water interface we employ the approach proposed by Chou and Fringer [7] to specify boundary conditions for \( \psi \) using van Rijn’s pickup function [39]. At rigid walls and the outlet of the flow domain we employ a Neumann, zero-gradient boundary condition while for the free-surface boundary we assume free-slip condition [39]. For details regarding the equations and their boundary condition we employ to model the various terms in the above equations see [22, 24].

Appendix 2: Model validation

The VFS-Geophysics model is extensively validated for morphodynamics computations against laboratory- and field-scale data [22,23,24]. Herein, we present the results of yet another validation study using the sediment transport data obtained in the OSL for the test case 3, which includes an installed longitudinal wall (smooth) an angle of attach of 40°. Validation studies for other test cases can be found in [23, 38]. As in the experiment, the numerical simulations started from flat sediment bed (see Fig. 10) and continued until the bed morphology reached a dynamic equilibrium. Once at the quasi-equilibrium, a subaqueous bed scanner was used to measure the bed topography of the OSL. These bed measurements represent instantaneous bed elevation of the OSL after \( t \sim 8 \) h during which OSL was at dynamic quasi-equilibrium, which can be characterized by the migration of mature bedforms. The amplitude of the experimentally observed bedforms was about 0.1–0.2 m while their wavelength varies between 0.5 and 1.0 m.

Fig. 10
figure 10

Initial bed elevations profiles at the beginning of the simulations and experiments (black dotted lines). The location of cross-sections is shown in legend

The initial bed topography of the OSL was scanned and used to create the computational grid system, which consists of \( \sim 9.9 \) million grids with \( 1201 \times 201 \times 41 \) nodes in streamwise, spanwise, and vertical directions, respectively. Given the vertical (\( \sim 0.3\,{\text{m}} \)), spanwise (\( \sim 2.5\,{\text{m}} \)), and longitudinal (\( \sim 15\,{\text{m}} \)) dimensions of the OSL, the spatial resolution of the grid, which is uniformly distributed, is about \( \sim 0.01\,{\text{m}} \). The initial bed profiles in the experiment and, consequently, the model are shown in Fig. 10, while in Fig. 11a we plot the details of the computational grid system. Turbulence is modeled using large-eddy simulation (LES) module of the VFS-Geophysics model [25, 26]. A time step of 0.1 s was utilized for the flow field calculations to ensure a CFL number of less than 1.0. At the water surface, we prescribed the measured water-surface elevation as a sloping-lid boundary condition. A uniform flow is prescribe at the inlet of the OSL, while wall-modeling approach is used at the bed and side-walls of the OSL. The inlet boundary condition for the sediment transport model was prescribed with an inlet flux of 2 kg/min, which is consistent with the sediment feeding rate in the experiment.

Fig. 11
figure 11

Computational mesh (a) and simulated morphodynamics (b, d) and hydrodynamics (c) of the flow and sediment transport in the OSL with a longitudinal wall at the outer bend of the meander. The angle of attack of main flow to the wall is 40°. a Shows the computational mesh used to simulate OSL with an installed wall. Sediment transport equation is solved on the unstructured red triangular mesh, the white structured mesh is the background mesh on which the flow field is solved. The white square background mesh is skipped by a factor of 5 for the sake of visual clarity. The black-dots are the points surveyed in the OSL to produce the geometrical data. The grid resolution is almost uniform in all directions with a spacing of about 1.5 cm. b Shows the contours of time-averaged bed elevation; c depicts the computed contours of instantaneous out-of-plane vorticity at the water surface; and d is the computed rms (percent) of bed elevation fluctuations. Flow is from left to right

We started the simulation by running the flow solver in the OSL under banckful flow conditions. Once the flow field was statistically converged (which was established by motoring the kinetic energy within the computational domain), the morphodynamics module was activated. The coupled flow and morphodynamics simulation was continued until the bed morphology of the channel reached its quasi-equilibrium. The computed instantaneous vorticity field of the OSL is shown in Fig. 11c. The simulated time-averaged bed elevation and the root–mean-square (rms) of bed elevation fluctuations are shown in Fig. 11b, d, respectively. The simulated bedforms migrate and, thus, the bed bathymetry continuously evolves. The time-averaged bed elevations are computed to produce a bed bathymetry that can be better compared with the measured bed profiles.

In Fig. 12 we compare the simulated and measured bed profiles along the seven cross-sections shown in Fig. 11b. One can clearly see a major discrepancy between measured and simulated bed profiles for the first two cross-sections. The reason for such discrepancies can be attributed to the uncertainty in the inlet boundary condition. We note that the measured bed profiles in Fig. 12 are instantaneous while the simulated results are time-averaged. Overall, the agreement observed in Fig. 12 for the cross-sections three to seven is reasonably good and the mean error percentage is about 10%. Additionally, as one can see in this figure, the generic scour and deposition areas are well predicted. Finally, we note that the computed rms of the bed fluctuation in Fig. 12 is a good representative of the amplitude of the numerically captured bedforms. The computed rms of bed fluctuation within the channel varies between 0.08 and 0.18 m with a maximum of about 0.19 m, which seems consistent with the amplitude of the experimentally observed bedforms in the OSL.

Fig. 12
figure 12

Computed time-averaged (red lines) and measured instantaneous (black circles) bed elevation (\( z \)) profiles along six cross-sections of test case 3 (shown in legend). The vertical axis on the right represents rms of bed fluctuations (red dotted-line) for the simulated bed morphology and \( S \) (in m) is the vector along each cross-section. Note that since the river banks are almost stationary, the rms of fluctuation near the river banks approaches zero. While in the mid-channel, where migrating bedforms are present in the simulations, rms values vary between 0.1 with a maximum of \( \sim 0.2 \)

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Khosronejad, A., Diplas, P., Angelidis, D. et al. Scour depth prediction at the base of longitudinal walls: a combined experimental, numerical, and field study. Environ Fluid Mech 20, 459–478 (2020). https://doi.org/10.1007/s10652-019-09704-x

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