Numerical analysis of seabed dynamic response in vicinity of mono-pile under wave-current loading

Jie Lin,Ji-sheng Zhang,Ke Sun,Xing-lin Wei,Ya-kun Guo*

aKey Laboratory of Coastal Disaster and Defence(Hohai University),Ministry of Education,Nanjing 210098,China

bCollege of Harbor,Coastal and Offshore Engineering,Hohai University,Nanjing 210098,China

cFaculty of Engineering&Informatics,University of Bradford,Bradford BD7 1DP,UK

Abstract

Keywords:Numerical simulation;Dynamic response;Wave-current loading;Mono-pile foundation;Porous seabed

1.Introduction

Mono-pile foundations play a key role in marine engineering,and have several distinct advantages,including high strength,resistance to bending,and high integrity.Such pile foundations are affected by wave-current loading in harsh offshore environments,threatening the stability of the structure.The foundations can be damaged under the load of a typhoon(Wei,2018).Due to its practical importance,extensive studies have been carried out to investigate the interaction between waves and offshore structures in past decades(Tao et al.,2018).Bennett(1977)measured the pore water pressure and hydrostatic pressure of silty clay in the Mississippi River Estuary and discovered the presence of excess pore water pressure.Maeno and Hasegawa(1985)proposed an empirical formula based on the second-order Stokes wave theory for the pore water pressure induced by waves.In their empirical formula,the distribution of pore water pressure from the surface to the bottom of the seabed was expressed with an equation governed by wave steepness and soil permeability.Seed and Idriss(1971)proposed a relationship between the pore water pressure within the seabed and the liquefaction depth.Jeng et al.(2001)established an approximate model of two-dimensional(2D)operation conditions based on Biot's consolidation equations to analyze the anisotropic seabed response.They found that the liquefaction depth increased with the wave steepness.Sui et al.(2016)investigated the wave-induced seabed response around a mono-pile foundation.Zhao et al.(2017)and Lin et al.(2017)developed numerical models to simulate the pore water pressure around the foundation generated by wave loading.Sui et al.(2017)studied the variation of soil around a mono-pile foundation based on a three-dimensional(3D)numerical model.The results showed that the probability of soil liquefaction around the pile was greatly increased due to the presence of the pile.

When waves and currents coexist,the interaction between currents and offshore structures becomes more complex.For example,Liu et al.(2007)conducted laboratory studies to show that the wave-current force on the pile increased with the wave height.Qi and Gao(2014)conducted experiments and showed that,under the action of wave-current forces,the response of pore water pressure in silt had a significant effect on the scouring around the pile.With the development of computational fluid dynamics and computing resources,various numerical models have been widely used as a costeffective method for investigating these abed response induced by wave-current loading.Duan et al.(2017)established a 2D coupled model to explore oscillatory non-cohesive soil liquefaction around the structure,and also built a 3D model.The results showed that the liquefaction depth of the seabed was related to the wave and current directions.

Currently,studies of structure responses under wavecurrent loading are still scarce,and this complex process is far from fully understood(Pu et al.,2019).The aforementioned studies mainly focused on the response of marine structures only under wave loading,ignoring the fact that the wave-current interaction would affect the dynamic response of the seabed.In this study,an integrated numerical model was used to accurately estimate responses of a seabed with a mono-pile foundation under nonlinear wave-current loading.In the wave-current model,Reynolds-averaged Navier-Stokes(RANS)equations were used to simulate the wave-current interactions.In the seabed model,Biot's consolidation equations were used to analyze the seabed response.The integration of the wave-current and seabed models to develop a 3D numerical model based on the continuity of the water pressure at the seabed surface is described in section 2.Section 3 presents the verification of the integrated model against available experimental data or the analytical solution.Section 4 describes the use of this integrated model to investigate the wave-current loading-induced dynamic response of the seabed around the mono-pile foundation.

2.Numerical model

In this study,the responses of a seabed with a mono-pile foundation under wave-current loading were investigated.Fig.1 shows the schematic diagram of a porous seabed with a half-buried mono-pile subject to wave-current loading,where l is the length of the seabed,d is the thickness of the seabed,D is the pile diameter,h is the water depth,and b is the buried depth.

Two numerical models,the wave-current model and seabed model,were introduced and integrated through the continuity of the water pressure at the seabed surface(Zhang et al.,2012,2014).Using the integrated model,a series of analyses of oscillatory liquefaction potential around the mono-pile foundation were conducted under the wave-current loading.

Fig.1.Sketch of seabed with mono-pile foundation under wave-current loading.

2.1.Wave-current model

Assuming that the fluid is non-compressible viscous fluid,the governing equations are the RANS equations:

where xiand xj(i,j=1,2,3)represent the Cartesian coordinate system; 〈ui〉and 〈uj〉are the mean velocity components in the xiand xjdirections,respectively;t is time;ρ is thefluid density;〈p〉is the fluid pressure;μ is the dynamic viscosity;and g is the gravitational acceleration.is the Reynolds stress term,which is obtained with the k-ε equation(Rodi,1993;Launder and Spalding,1974;Zheng et al.,2014).Based on the eddy-viscosity assumption,the Reynolds stress term can be estimated as

where μtis the turbulent viscosity,k is the turbulence kinetic energy,and δijis the Kronecker delta.

Eq.(2)can then be written as

where μeffis the total effective viscosity.

The standard equations for the k-ε model can be written as follows(Rodi,1993):

where ε is the dissipation rate of turbulence kinetic energy,and Pkis the Prandtl number for turbulent transport.The constant values used in this model were Cμ=0.09,σk=1.00,σε=1.30,Cε1=1.44,and Cε2=1.92(Rodi,1993).

The volume of fluid(VOF)method(Hirt and Nichols,1981)was used to track the free water surface,using a function F to represent the fractional volume of water:

where uiis the velocity component in the xidirection.In Eq.(8),F=1 indicates that the mesh is full of water,F=0 indicates that the mesh is filled with air,and 0＜F＜ 1 means the existence of a free-water surface.

2.2.Seabed model

Biot's consolidation equations were used to integrate the soil-fluid interaction between two-phase media in the seabed model(Biot,1941).The seabed was considered an elastic material,and the pile was considered a rigid body that was completely impervious to water.

Based on Biot's consolidation equations,the governing equation of force balance can be expressed as

The mass conservation equation can then be obtained:

where nsis the porosity of the soil,and ksis the soil permeability that was constant in the seabed in this study.The fluid compressibility β and volumetric strain εscan be calculated by(Kriebel,1998)

where us,vs,and wsare the soil displacements in the x,y,and z directions,respectively;Sris the degree of saturation;Pw0is the absolute static water pressure;and Kfis the bulk modulus of the pore water,which is usually taken to be 2.24×109N/m(Yamamoto et al.,1978).

According to the generalized Hooke's law,the relationship between the incremental elastic stress and the soil displacement can be determined according to the following equations:

where μsis the Poisson's ratio;and G is the shear modulus,with G=Es/［2（1+ μs）］,where Esis the Young's modulus.

2.3.Boundary conditions

2.3.1.Wave-current boundary conditions

In the 3D numerical wave-current flume established in this study,the front end of the flume was set as the wave boundary for wave and current generation.The pressure boundary was applied at the tail end of the flume,and the bottom of the flume was specified as a non-slip wall boundary.At the top of the flume,the atmospheric pressure boundary was employed,and the relative pressure at the water surface was set to zero.The other surfaces were set to the symmetric boundaries.

2.3.2.Seabed boundary conditions

To better simulate the wave-current field,appropriate seabed boundary conditions are needed.On the surface of the seabed,the pore water pressure(ue)was assumed to be equal to the wave pressure(Pb).Meanwhile,because the effective stress in the vertical direction was less than 2% of the wave pressure on the surface of the seabed,it was ignored in this study.

The shear stress on the seabed surface was considered to be caused by the current,i.e.,the shear stress on the walls could be expressed as,where u*is the frictional velocity.However,owing to the high wave pressure used in this study,the frictional stress at the bottom was small and could be ignored.Therefore,the assumption could be simplified to

Since the soil was on the impervious rigid foundation,there was no displacement and vertical flow at the junction of the soil-rigid bottom,where z=-d,which can be expressed as

In the seabed model,the lateral and bottom boundaries were regarded as impervious boundaries.The values of the pore water pressure and soil displacement at the boundary were assumed to be zero.At the interface between the wave and structure,the loading on the structure was regarded as the sum of the wave pressure and the shear stress generated by currents(Biot,1941).The effects of air and other media on the structure were less than the effects of wave-current loading,which were not considered in this study.

2.4.Model integration

To simplify the calculation,the seabed simulation area was determined to be not smaller than 2L×L×L(Ye and Jeng,2012),where L is the wavelength.The mono-pile foundation was set at the center of computational domain.Therefore,in the present model,the length of the seabed was set as 200 m.The displacement of the mono-pile foundation changed slightly and its influence on the propagation of waves and currents was negligible.The continuity of the hydrodynamic pressure was used to integrate the wave-current model and the seabed model.Correspondingly,a 3D integrated model was established,with dimensions of 10L×5L×2h,where h is the water depth.

In general,the mesh size of the wave-current model is smaller than that of the seabed model.In the wave-current model,the whole region was discretized into 2.4×106quadrilateral meshes,with the size of H/100 = 0.07 m,where H is the wave height(Zhao,2010).With the finite difference method,it is concluded that a stable calculation result can be obtained by dividing the wave period into 40 steps.In the seabed model,the seabed consisted of 269 222 quadrilateral elements with a size of 0.7 m,and the model was solved with the finite element method.

In the simulation,the MATLAB interface of the model was used for iterative calculation(Zhao et al.,2017).A wave period T was divided into n computational time steps.In thefirst computational time step,the wave pressure at the initial time was taken as the boundary condition of the seabed and structure surfaces to calculate the seabed response during Δt=T/n.The calculation results obtained in the previous time step were then used as the initial values of the next time step with the corresponding wave pressure applied to the seabed.By analogy,the cyclic loading of wave pressure was carried out using the MATLAB script.

3.Numerical model validation

Model validation was conducted for two simple cases:(1)the wave-current interaction,and(2)the interaction between waves and the seabed.

Simulated water surface elevations Z induced by wavecurrent interaction were compared with the laboratory experimental data and the results of the third-order Stokes wave theory for the case without piles.The wave-current model was simplified to ensure that it was consistent with the results of Umeyama(2010).The input data are shown in Table 1.Fig.2 shows that the simulated water surface elevation agrees with the laboratory measurement.Some deviations between the simulation and measurement occurred,which was mainly caused by the simulation of higher-order nonlinear waves.

The integrated model was validated by comparison of its results with laboratory experimental data of Liu et al.(2015),in which the pore water pressure was measured at depths of z=-6.7 cm and z=-26.7 cm.The parameters used for model validation are listed in Table 2.Fig.3 shows the comparison of the simulated and measured pore water pressures.In general,the simulated pore pressures agree with the measurements.

4.Numerical results and discussion

The validated numerical model was used to study the seabed dynamic response in the vicinity of a mono-pile foundation under the wave-current loading.To explore the factors that may have a significant impact on the dynamic response,the wave pressure obtained from the wave-current model was imposed as the boundary condition of the seabed model for simulating the distributions of the pore water pressure,effective stress,and liquefaction depth.

The wave-current characteristics and properties of the seabed and mono-pile are listed in Table 3.A negative value of current velocity indicates that the current travels against the wave.

The seabed was considered an isotropic homogeneous media.The simulated area of the seabed,with a length of 200 m,a width of 41 m,and a thickness of 40 m,is shown in Fig.4.The pile,with a diameter of 1.4 m and a buried depth of21 m,was placed at the central position of the porous seabed along the x-axis.Three typical positions,A,B,and C,were selected for investigating the physical process around the mono-pile foundation,where A and C were located in front of and at the leeward side of the mono-pile following the wave propagation direction,and B was on the lateral side of the mono-pile.

Table 1Parameters used for verification of wave-current model.

Fig.2.Comparison of simulated water surface elevations with measurements under wave-current loading.

Table 2Parameters used for validation of integrated model.

Fig.3.Comparison of simulated and measured pore water pressures.

4.1.Analysis of pore water pressure for different current velocities

Fig.5 shows the distribution of the pore water pressure on the seabed at the wave trough with different current velocities.In general,the pore water pressure has its maximum value at the seabed surface.Fig.5 shows that the magnitude of the pore water pressure decreases rapidly from the seabed surface to the upper layer,and then slightly decreases with the increase of the depth to the bottom of the seabed.In the case when the current traveled against the wave(Fig.5(a)and(b)),the pore water pressure decreased with the increase of the current velocity.This is because the wave-current interaction changed the original wave height and wavelength.When the mono-pile was not involved in the study,the wavelength was elongated and the wave height was reduced by the current traveling in the direction of wave propagation.When a mono-pile was embedded into the seabed,the wave height increased due to the blockage of the mono-pile,which might further increase the wave pressure in the seabed(Duan et al.,2019).As shown in Fig.5(d)and(e),the presence of forward flow increases thepore water pressure in the seabed,while the presence of a countercurrent results in the opposite.

Table 3Parameters of wave-current,seabed,and mono-pile.

Fig.4.Schematic diagram of test positions.

Fig.5.Distributions of wave-current loading-induced pore water pressures in seabed for different current velocities.

4.2.Analysis of effective stress for different current velocities

Waves propagating over the seabed surface generate periodical loading on the seabed and lead to the variation of the effective stresses in the porous media.Due to the large seepage force occurring on the seabed at the wave trough,the structure suffers the maximum wave force and produces the maximum effective stress.Therefore,the changes of the effective stress along three vertical lines at positions of A,B,and C at the wave trough were explored.

4.2.1.Effective stress along vertical line A-A′

The effective stresses in each direction along the vertical line A-A′are shown in Fig.6.Despite the variations in both current velocity and direction,the trends of the curves in the x direction are almost the same for different current velocities.

Fig.6(a)shows that the maximum value ofappears at the seabed surface.Taking the cases of U0=-1.0 m/s and U0=1.0 m/s as examples,the maximum values ofwere about 14 kPa and 43 kPa at the seabed surface.Fig.6(b)shows thatalong the vertical line A-A′has a similar trend tocurves along the vertical line A-A′are generally consistent at different current velocities.As shown in Fig.6(c),there is a mutation of seabed response near the seabed surface,which is different from Fig.6(a)and(b).This abnormal result of discontinuity was shown in Madsen(1978).The peak value ofoccurs at the depth of about-3 m,due to the fact that a horizontal seepage force exists in the soil between the locations corresponding to the wave crest and wave trough.It can be concluded from Fig.6(c)that the direction of wave propagation over the current may largely affect the variation amplitude ofand its value at the same depth.

Fig.6. ,,andalong vertical line A-A′for various current velocities.

4.2.2.Effective stress along vertical line B-B′

Fig.7 shows the vertical distributions of the effective stress along B-B′for the depth ranging from 0 to-20 m.When only waves exist,shows its maximum value at the seabed surface.Then,the curve approaches 0 when it plunges to-3 m,as shown in Fig.7(a).When the directions of waves and currents are reversed,shows its minimum value at the seabed surface,which is different from that at position A.The amplitude ofand its change trend along B-B′are the same as those of,and the change trend ofalong B-B′,shown in Fig.7(c),is consistent with that in Fig.6(c).

4.2.3.Effective stress along vertical line C-C′

Effective stresses in the x,y,and z directions along the vertical line C-C′are presented in Fig.8.When the current travels in the opposite direction of the wave propagation,the seabed surface hasthe smallestnegative value of(Fig.8(a)).The value increases sharply to 0 with the increase of the seabed depth,which is consistent with the change trend at point B.In the case with only the wave movement,the minimum stress value occurs in the surface layer,which is inconsistent with that at positions A and B.

4.3.Wave-current loading-induced liquefaction around mono-pile

Liquefaction is defined as the state of soil in which the average effective stress drops to zero,and the water-sediment mixture loses its carrying capacity,leading to seabed instability and the destruction of the marine structure(Sumer,2014).The criterion for determining the liquefaction of soil in a 3D space is as follows(Tsai,1995):

where ρsis the density of the submerged soil;and K0is the coefficient of lateral earth pressure at rest,which can be determined as

According to Eq.(23),the criterion for instantaneous liquefaction can be obtained:

Fig.7. ,,and along vertical line B-B′for various current velocities.

Fig.8. ,,andalong vertical line C-C ′for various current velocities.

Fig.9.Instantaneous liquefaction depth distributions around mono-pile with various current velocities.

where σ′is the effective stress,with）/3;and σ0is obtained from the density of the submerged soil,and σ0= ρsgz（1+2K0）/3.

Based on the liquefaction criterion,the results of the instantaneous soil liquefaction around the mono-pile under various current velocities are shown in Fig.9.Fig.9(a)and(b)shows that,at the front and side positions,liquefaction does not occur through the whole depth of the seabed.As shown in Fig.9(c),liquefaction occurs downstream of the pile.However,the upper layer of the seabed is liquefied whether the wave and current travel in the same direction or not.The current traveling against the wave results in a larger momentary liquefaction depth of the seabed.For example,the seabed liquefaction depth reached-2 m when the current velocity was 1 m/s.However,when the current velocity was-1 m/s,the seabed liquefaction depth reached-5.5 m.Hsu et al.(2009)showed that the increase of the current velocity increased the liquefaction depth.In their study,a potential flow theory was adopted,and the current was assumed to be uniform,which was contrary to the fact that the current velocity decreases near the sea bottom due to the non-slip boundary.It can be concluded that the computational fluid dynamics(CFD)method,combing RANS equations with the k-ε turbulence model,is suitable for simulating the wave-current interaction as it considers more realistic situations than the potential flow theory.

Results show that the positions in front of and on the side of the mono-pile are relatively safe,compared with the position downstream of the pile.The seabed surface downstream of the pile is prone to be liquefied,which can be ascribed to the fact that negative values ofwill lead to the liquefaction of the seabed.To protect the seabed around the mono-pile against instantaneous soil liquefaction,corresponding protective measures should be taken downstream of the pile foundation.

5.Conclusions

The pore water pressure in the seabed and the variation of the effective stress around the mono-pile under wave-current loading were investigated for various current velocities.The following conclusions are drawn:

(1)When the current travels along the wave,the shear stress in the seabed will increase,leading to an increase of the pore water pressure;when the current travels against the wave,the pore water pressure will decrease.

(2)Whether the wave and current travel in the same direction or not,the effective stresses in the x and y directions are the largest at the seabed surface,and the value ofdecreases with the value of the current velocity at the same depth of the seabed.There are mutations in thecurves.The value ofat the same depth of the seabed gets smaller for a larger value of the current velocity in front of and on the side of the mono-pile,while the opposite situation occurs at the leeward side of the mono-pile.

(3)The current traveling along the wave decreases the liquefaction depth of the porous seabed by about 50%,as compared to the condition under which the current travels against the wave,with a current velocity of 1 m/s in this study.