Effect of bed roughness on turbulent boundary layer and net sediment transport under asymmetric waves

This paper presents an investigation of the roughness effects in the turbulent boundary layer for asymmetric waves by using the baseline (BSL) k–ω model. This model is validated by a set of the experimental data with different wave non-linearity index, N_i (namely, N_i = 0.67, N_i = 0.60 and N_i = 0.58). It is further used to simulate asymmetric wave velocity flows with several values of the roughness parameter (a_m/k_s) which increase gradually, namely from a_m/k_s = 35 to a_m/k_s = 963. The effect of the roughness tends to increase the turbulent kinetic energy and to decrease the mean velocity distribution in the inner boundary layer, whereas in the outer boundary layer, the roughness alters the turbulent kinetic energy and the mean velocity distribution is relatively unaffected. A new simple calculation method of bottom shear stress based on incorporating velocity and acceleration terms is proposed and applied into the calculation of the rate of bed-load transport induced by asymmetric waves. And further, the effect of bed roughness on the bottom shear stress and bed-load sediment transport under asymmetric waves is examined with the turbulent model, the newly proposed method, and the existing calculation method. It is found that the higher roughness elements increase the magnitude of bottom shear stress along a wave cycle and consequently, the potential net sediment transport rate. Moreover, the wave non-linearity also shows a big impact on the bottom shear stress and the net sediment transport.     

Numerical simulation of wave-induced laminar boundary layers

The three-dimensional numerical model with σ-coordinate transformation in the vertical direction is applied to the simulation of surface water waves and wave-induced laminar boundary layers. Unlike most of the previous investigations that solved the simplified one-dimensional boundary layer equation of motion and neglected the interaction between boundary layer and outside flow, the present model solves the full Navier–Stokes equations (NSE) in the entire domain from bottom to free surface. A non-uniform mesh system is used in the vertical direction to resolve the thin boundary layer. Linear wave, Stokes wave, cnoidal wave and solitary wave are considered. The numerical results are compared to analytical solutions and available experimental data. The numerical results agree favorably to all of the experimental data. It is found that the analytical solutions are accurate for both linear wave and Stokes wave but inadequate for cnoidal wave or solitary wave. The possible reason is that the existing analytical solutions for cnoidal and solitary waves adopt the first-order approximation for free stream velocity and thus overestimate the near bottom velocity. Besides velocity, the present model also provides accurate results for wave-induced bed shear stress.     

Flow under standing waves: Part 1. Shear stress distribution,energy flux and steady streaming

The conditions for energy flux, momentum flux and the resulting streaming velocity are analysed for standing waves formed in front of a fully reflecting wall. The exchange of energy between the outer wave motion and the near bed oscillatory boundary layer is considered, determining the horizontal energy flux inside and outside the boundary layer. The momentum balance, the mean shear stress and the resulting time averaged streaming velocities are determined. For a laminar bed boundary layer the analysis of the wave drift gives results similar to the original work of Longuet–Higgins from 1953. The work is extended to turbulent bed boundary layers by application of a numerical model. The similarities and differences between laminar and turbulent flow conditions are discussed, and quantitative results for the magnitude of the mean shear stress and drift velocity are presented. Full two-dimensional simulations of standing waves have also been made by application of a general purpose Navier–Stokes solver. The results agree well with those obtained by the boundary layer analysis. Wave reflection from a plane sloping wall is also investigated by using the same numerical model and by physical laboratory experiments. The phase shift of the reflected wave train is compared with theoretical and empirical models.     

Numerical simulation of a partially buried pipeline in a permeable seabed subject to combined oscillatory flow and steady current

Hydrodynamic forces exerting on a pipeline partially buried in a permeable seabed subjected to combined oscillatory flow and steady current are investigated numerically. Two-dimensional Reynolds-Averaged Navier-Stokes equations with a k−ω turbulent model closure are solved to simulate the flow around the pipeline. The Laplace equation is solved to calculate the pore pressure below the seabed with the simulated seabed hydrodynamic pressure as boundary conditions. The numerical model is validated against the experimental data of a fully exposed pipeline resting on a plane boundary under various flow conditions. Then the flow with different embedment depths, steady current ratios and KC numbers is simulated. The amplitude of seepage velocity is much smaller than the amplitude of free stream velocity as expected. The normalized Morison inertia, drag and lift coefficients based on the corresponding force coefficients of a fully exposed pipeline are investigated. The normalized Morison force coefficients reduce almost linearly with the increase of embedment depth and that the KC only has minor effect on the normalized Morison coefficients. It is also found that the permeable seabed condition causes a slight increase on the inline force and has a little effect on the lift force, compared with corresponding conditions in an impermeable bed.     

Turbulence-induced steady streaming in an oscillating boundary layer: On the reliability of turbulence closure models

The accuracy of several closure models of the Reynolds-Averaged Navier–Stokes Equations in predicting the characteristics of an oscillating turbulent wall boundary layer is analyzed. The analysis involves four low Reynolds number k − ε models and a k − ω model and it is carried out by comparing the model results both with experimental data and with data obtained by a Direct Numerical Simulation (DNS) of the Navier–Stokes equations. The boundary layer is generated by a spatially constant time-oscillating pressure gradient given by the sum of two harmonic components characterized by angular frequencies Ω^∗ and 2Ω^∗ respectively, which generates a steady streaming because of the asymmetry of turbulence intensity during the cycle. Thus the results are relevant to the boundary layer at the bottom of nonlinear sea waves. The attention is therefore focused on the accuracy of the models in reproducing the period averaged profiles of the hydrodynamic characteristics of the steady streaming. The instantaneous quantities, such as time development of the wall shear stress, profiles of the streamwise velocity, Reynolds stresses and turbulent kinetic energy are also considered and analyzed. The results shows that a model can be judged better or worse than other models depending on the specific flow characteristic under investigation. However, an approach has been adopted which allowed to rank the models according to their accuracy in predicting the values of the hydrodynamic quantities involved in the present study.     

Numerical simulation of wave transformation and runup incorporating porous media wave absorber and turbulence models

A numerical wave tank is first established using the Navier–Stokes equations and the VOF method assuming laminar flow. The standard k–ε, realizable k–ε and RNG k–ε turbulent models are then incorporated to the numerical tank. An effective numerical method for wave absorption utilizing the energy-dissipating property of porous media is also included. To validate the accuracy of the proposed models, the propagation of a solitary wave, where analytical solution is available for comparison, is first simulated. This is followed by the simulation of irregular wave runup on a composite seawall, wave propagation over submerged bars and wave refraction and diffraction over an elliptic shoal, where experimental data are available for comparison. All computed results agree well with either the analytical solution or the experimental data.     

Bed friction and dissipation in a combined current and wave motion

Two simple two-layer eddy viscosity models, which facilitate analytical solutions, are presented in order to describe the velocity field and associated shear stress in a combined current wave motion. The models, which have the same eddy viscosity in the current boundary layer, but different eddy viscosities in the wave boundary layer, cover together the whole rough turbulent regime. Straightforward definitions are made for the wave friction factor and the current friction factor for the combined motion, which are in accordance with the results for pure waves and pure currents. In this way one avoids the fictitious reference velocities and elliptic integrals which e.g. Grant and Madsen (1978, 1979) experienced. The two friction factors turn out to be functions of four dimensionless parameters. A detailed calculation procedure is presented. Comparison with laboratory experiments yields promising results. A new relation connecting dissipation and bed shear stress is also developed.     

Numerical modeling of flow and scour below a pipeline in currents: Part I. Flow simulation

The performance of the standard k–ε, Wilcox high-Reynolds-number k–ω, Wilcox low-Reynolds-number k–ω and Smagorinsky's subgrid scale (SGS) turbulence models is examined against the flow around a circular cylinder 0.37 diameter above a rigid wall. The governing equations are solved using finite difference method in a non-orthogonal boundary-fitted curvilinear coordinate system. A mesh dependence study for the four turbulence models is carried out on computational meshes with different densities. In addition, the performance of the k–ω models with either wall function or no-slip boundary condition on the cylinder surface is examined on the finest mesh. It is found that the SGS model over-predicts the shedding of vortices from the cylinder and is sensitive to the computational mesh and the model constant C_s used. The standard k–ε and the Wilcox k–ω models predict the mean velocity field quite well but generally under-predict the velocity and hydrodynamic force oscillations using wall functions on the cylinder surface. It is also found that the Wilcox k–ω models with the no-slip boundary condition on the cylinder surface give better predictions on the shedding of vortices than their counterparts using the wall function boundary condition.     

A model of oscillatory rough turbulent boundary layer flow

Approximate analytical solutions of the boundary layer equation are obtained in closed form for oscillatory rough turbulent flow. The solutions are based on a time-varying eddy viscosity, and the aim of the study is to assess the effects of these time variations on the properties of the wave boundary layer. The flow and the eddy viscosity are made interdependent by a closure assumption which relates the peak value of viscosity in the wave cycle to the peak value of shear velocity. Instantaneous vertical profiles of horizontal velocity and shear stress, and time series of the bed shear stress, are presented for a typical case study. In addition, the wave drag coefficient, the boundary layer thickness and the phase lead of peak bed shear stress over peak free-stream velocity, are determined as functions of both the relative roughness and the parameter governing the magnitude of the time variations in viscosity. Reasonable agreement is demonstrated with previous experimental and theoretical results.     

Extending the k–ω turbulence model towards oceanic applications

Four different two-equation turbulence models for geophysical flows are compared: The k–ϵ model, two new versions of the k–ω model, and the Mellor–Yamada model. An extension of the k–ω model for buoyancy affected and rotating flows is suggested. Model performance is evaluated for a few typical oceanic flows. First, new analytical solutions of the models for the surface layer affected by breaking surface waves are discussed. The deficiencies of earlier attempts are high-lighted, and it is demonstrated why the Mellor–Yamada model and the k–ϵ model fail. It is illustrated that only one version of the k–ω model computes correct decay rates for turbulent quantities under breaking waves. Second, it is demonstrated that all models predict almost identical mixed layer depths and profiles for the turbulent kinetic energy in a classical stratified shear-entrainment experiment if the buoyancy term in the second equation is appropriately weighted. Third, the accuracy and numerical robustness of the new k–ω model in realistic oceanic situations is confirmed by comparison with the data-set of the Ocean Weather Ship ‘Papa’.     

Seabed shear stress and bedload transport due to asymmetric and skewed waves

A simple conceptual formulation to compute seabed shear stress due to asymmetric and skewed waves is presented. This formulation generalizes the sinusoidal wave case and uses a variable friction factor to describe the physics of the boundary layer and to parameterize the effects of wave shape. Predictions of bed shear stresses agree with numerical computations using a standard boundary layer model with a k–ε turbulence closure. The bed shear stress formulation is combined with a Meyer-Peter and Müller-type formula to predict sheet flow bedload transport under asymmetric and skewed waves for a horizontal or sloping bed. The predictions agree with oscillatory water tunnel measurements from the literature.     

Numerical modeling investigation on turbulent oscillatory flow over a plane rough bed composed by randomly arrayed particles

A three-dimensional numerical model is established to simulate the turbulent oscillatory boundary layer over a fixed and rough bed composed by randomly arrayed solid spheres based on the lattice Boltzmann method and the large eddy simulation model.The equivalent roughness height,the location of the theoretical bed and the time variation of the friction velocity are investigated using the log-fit method.The time series of turbulent intensity and Reynolds stress are also investigated.The equivalent roughness height of cases with Reynolds numbers of 1×10~4–6×10~4 is approximately 2.81 d(grain size).The time variation of the friction velocity in an oscillatory cycle exhibits sinusoidal-like behavior.The friction factor depends on the relative roughness in the rough turbulent regime,and the pattern of solid particles arrayed as the rough bed in the numerical simulations has no obvious effect on the friction factor.     

Laboratory observation of boundary layer flow under spilling breakers in surf zone using particle image velocimetry

A boundary layer flow under spilling breakers in a laboratory surf zone with a smooth bottom is investigated using a high resolution particle image velocimetry (PIV) technique. By cross-correlating the images, oscillatory velocity profiles within a viscous boundary layer of O(1) mm in thickness are resolved over ten points. Using PIV measurements taken for an earlier study and the present study, flow properties in the wave bottom boundary layer (WBBL) over the laboratory surf zone are obtained, including the mean velocities, turbulence intensity, Reynolds stresses, and intermittency of coherent events. The data are then used to estimate the boundary layer thickness, phase variation, and bottom shear stress. It is found that while the time averaged mass transport inside the WBBL is onshore in the outer surf zone, it changes to offshore in the inner surf zone. The zero Eulerian mass transport occurs at h/h_b ≈ 0.92 in the outer surf zone. The maximum overshoot of the streamwise velocity and boundary layer thickness are not constant across the surf zone. The bottom shear stress is mainly contributed by the viscous stress through mean velocity gradient while the Reynolds stress is small and negligible. The turbulence level is higher in the inner surf zone than that in the outer surf zone, although only a slight increase of turbulent intensity is observed inside the WBBL from the outer surf zone to the inner surf zone. The variation of phase inside and outside the WBBL was examined through the spatial velocity distribution. It is found the phase lead is not constant and its value is significantly smaller than previous thought. By analyzing instantaneous velocity and vorticity fields, a remarkable number of intermittent turbulent eddies are observed to penetrate into the WBBL in the inner surf zone. The size of the observed large eddies is about 0.11 to 0.16 times the local water depth. Its energy spectra follow the − 5/3 slope in the inertial subrange and decay exponentially in the dissipation subrange.     

Characteristics of the boundary layer at the bottom of a solitary wave

The results of direct numerical simulations of the boundary layer generated at the bottom of a solitary wave are described. The numerical results, which agree with the laboratory measurements of Sumer et al. (2010) show that the flow regime in the boundary layer can be laminar, laminar with coherent vortices and turbulent. The average velocity and bottom shear stress are computed and the results obtained show that the logarithmic law can approximate the velocity profile only in a restricted range of the parameters and at particular phases of the wave cycle. Moreover, the maximum value of the bottom shear stress is found to depend on the dimensionless wave height only, while the minimum (negative) value depends also on the dimensionless boundary layer thickness. Diagrams and simple formulae are proposed to evaluate the minimum and maximum bottom shear stresses and their phase shift with respect to the wave crest.     

Observations of turbulence under weakly and highly stratified conditions in the Ariake Sea

Observations of turbulence, stratification, and mean current were made using a microstructure profiler and an acoustic Doppler current profiler (ADCP) during four cruises at a central location in the Ariake Sea, under weakly and strongly stratified conditions. Continuous measurements of the dissipation rate of turbulent kinetic energy (TKE), ε, were made. These revealed that frictional bed turbulence with quarterdiurnal variation in the bottom boundary layer (BBL) was one of the most energetic sources of vertical mixing in the sea. Thickness of the BBL was strongly confined by the stable stratification. We investigate a relationship between the BBL height h and the Ozmidov scale. We present a systematic argument that describes the vertical structure and characteristic scales of velocity and turbulence inside the frictional BBL, where the stratification persisted. Considerable deviation of observed vertical shear from the law of the wall indicated a modification of turbulent scales by the stratification. Shear stress calculated from the velocity data using vertical integration of the equation of motion was found to decrease approximately linearly with height. The TKE production rate P, estimated using the shear stress, was highly correlated with the dissipation rate. The buoyancy contribution to TKE balance in the BBL was quantified in terms of the flux Richardson number R _f as R _f?=?0.12.     

Simulation of a 3D submerged jet flow around a pile

The submerged 3D turbulent jet flow behavior around a pile on a rigid bed and on a scoured bed was studied experimentally and numerically. ADV was used to obtain the jet velocity distributions and Realizable k–ε turbulence model was used for the prediction of flow field around a pile. The jet flow area was three-dimensional and thus numerical model was a three-dimensional model. The numerical results were used to predict the velocity close to the pile and bed shear stress on the bed. In general, the results indicated that the flow field was also in agreement with the findings of previous experiments in literature and the related principles in the subject area. The experimental results demonstrated that Acoustic Doppler Velocimeter (ADV) measurements were almost identical with the Realizable k–ε turbulence model results for turbulence intensity I=10%.     

Numerical computations of resonant sloshing using the modified isoAdvector method and the buoyancy-modified turbulence closure model

Sloshing is an interfacial-flow phenomenon which brings two challenges on how to locate the position of the interface and avoid the unphysical motion of the interface. In order to locate the the position of the interface, a new geometric Volume-of-Fluid (VOF) method called isoAdvector is adopted to pursue a sharp interface. Aiming to make the isoAdvector method compatible with the dynamic mesh adopted to handle the tank motion, the motion-flux correction is introduced, and a moving-velocity correction for face-interface intersection line (FIIL) is proposed. An approximation formula is adopted to effectively reconstruct the moving-velocity field of the meshes at each cell center based on the motion fluxes on each cell face. In order to avoid the unphysical motion of the interface due to the excessive turbulence level in the transition region at the interface, the buoyancy-modified k − ω SST model is adopted. The numerical results of wave elevations and forces are compared with the experiments. The comparisons suggest that (i) the moving-velocity correction for FIIL is important to update the volume fraction; (ii) the modified isoAdvector method can capture the the position of the interface more accurately than the algebraic VOF method; (iii) the unphysical motion of the interface can be avoided by using the buoyancy-modified k − ω SST model in long-time simulations. In addition, a new post-processing approach is proposed to evaluate the interface thickness. The decrease of interface thickness improves the accuracies of wave elevations by using the modified isoAdvector method. The adoption of both the modified isoAdvector method and the buoyancy-modified k − ω SST model improves the computational accuracies of wave elevations and hydrodynamic loads in long-time simulations.     

A Lagrangian model for boundary layer growth and bed shear stress in the swash zone

A new model for the boundary layer development and associated skin friction coefficients and shear stress within the swash zone is presented. The model is developed within a Lagrangian reference frame, following fluid trajectories, and can be applied to both laminar flow and smooth turbulent flow. The model is based on the momentum integral approach for steady, flat-plate boundary layers, with appropriate modifications to account for the unsteady flow regime and flow history. The model results are consistent with previous measurements of bed shear stress and skin friction coefficients within the swash zone. These indicate strong temporal and spatial variation throughout the swash cycle, and a clear distinction between the uprush and backwash phase. This variation has been previously attributed the unsteady flow regime and flow history effects, both of which are accounted for in the new model. Fluid particle trajectories and velocity are computed using the non-linear shallow water wave equations and the boundary layer growth across the entire swash zone is estimated. Predictions of the bed shear stress and skin friction coefficients agree reasonably well with direct bed shear stress measurements reported by Barnes et al. (Barnes, M.P., O’Donaghue, T., Alsina, J.M., Baldock, T.E., 2009. Direct bed shear stress measurements in bore-driven swash. Coastal Engineering 56 (8), 853–867) and, for a given flow velocity, give stresses which are consistent with the bias toward uprush sediment transport which has consistently been observed in measurements. The data and modelling suggest that the backwash boundary layer is initially laminar, which results in the late development of significant bed shear during the backwash, with a transition to a turbulent boundary layer later in the backwash. A new conceptual model for the boundary layer structure at the leading edge of the swash is proposed, which accounts for both the no-slip condition at the bed and the moving wet–dry interface. However, further development of the Lagrangian Boundary Layer Model is required in order to include bore-generated turbulence and to account for variable roughness and mobile beds.