Jing-xin Zhang,Jian Wang, Xiang Fan,Dongfang Liang

1.Key Laboratory of Hydrodynamics(Ministry of Education),shanghai Jiao Tong University,Shanghai 200240,China

2.School of Naval Architecture,Ocean and Civil Engineering,Shanghai Jiao Tong University,Shanghai 200240,China

Abstract:The hydrodynamics in a straight open channel with a multiple-embayment groyne field was investigated using the detached-eddy simulation (DES).A series of short groynes were included on a 1:3 side slope of the channel.This work focuseson the turbulent coherent structures around groynes on an uneven bottom. Flows around groynefieldsare characterized by massive separation and highly unsteady vortices.DEScan capture a wide spectrum of eddies at a lower computational cost than the large eddy simulation (LES)or direct numerical simulation (DNS).In the present work,a zonal DESmodel (ZDES)was used to simulate the flow around groynes. The ZDES model is a modified version of the DES designed to overcome the model-stress depletion (MSD) of the RANS/LES hybrid model. The vortex system consists of the horseshoe vortex (HV) formed at the base of the obstructions, the necklace vortex (NV)that wrapped the groyne tips near the free surface,and theshedding vortex (SV) underneath the freesurface.The effects of the incident flow and local topography on the vortex evolution were investigated by analyzing the mean flow structures and the instantaneous turbulent flow fields.Some important vorticescannot becaptured because of theaveraging process, while some flow structurescannot be observed in the instantaneous flow.The mean flow is only a reflection of the averaging process when complex vortices are present.

Key words:Open channel flow,groyne,free surface,detached eddy simulation, turbulent coherent structure

Introduction

Groynes are commonly built in rivers at an angle to the main flow to fulfil multiple objectives.One of the most common objectives is to maintain a channel’s navigability by keeping the flow away from the banks,increasing the flow velocity in the channel and encouraging sediment transportation.There are other objectives as well,e.g.,restoring fish habitats by degrading the river current. One characteristic of the groyne field flow is the considerably low velocity in the embayment region compared with the main flow.Because the embayment region between successive groynes acts as a dead water zone,the fluid and dissolved mass exchange ratio decreases,and the residence times of particulate matter are much longer,which may raise an eco-hydraulic problem[1].

The understanding of the hydrodynamics and the overall mass exchange between the embayment and the main channel are very important.Quantitative knowledge of the 3-D turbulent coherent structures and their dynamics will facilitate a better insight into the dynamic process.The turbulence-coherent structures are dominated by channel geometry and groyne shapes,including groyne orientation,spacing of the embayment,whether it is emerged,submerged,or permeable,and groyne tip shape.Most of the knowledge about the dynamics of turbulent flows around groyne fields comes from scale-model experimental researches.Using particle tracking velocimetry (PTV) techniques to capture the mean velocity and instantaneous turbulence quantities is an efficient laboratory methodology.The influencing parameters(e.g.,different orientation,emerged or submerged,different bottom geometries)have been thoroughly investigated[2-6].All these studies focused on the free surface flow patterns,mixing layer evolution,and mass exchange between the embayment and the main channel.Besides the PTV measurements,a 3-D acoustic Doppler velocimeter (ADV)was used to obtain the temporal velocities at certain locations.The advantage of ADV is in analysing the turbulent fluctuations and the turbulent kinetic energy (TKE),however,the extraction of flow information from only a single point during the experiments is often insufficient and is the obvious disadvantage compared with PTV or PIV measurements.

Compared to a physical model,a numerical model is much better at capturing the spatial and temporal characteristics of complex turbulent flow.The early numerical simulations were based on the RANS models.The understanding of the turbulent flow using RANSmodels is insufficient because some important vortices cannot be successfully captured due to the averaging process.Uijttewaal and van Schijndel[7]performed a 2-D depth-integrated large eddy simulation (LES)model,which was used to investigate horizontal turbulent coherent structures.The 3-D LES simulations were used to further investigate the turbulent coherent structures and seek physical mechanisms of the flow separation[8-10].The LES is able to capture unsteady vortices with multiscales,but the higher computational cost hinders its practical application.As one kind of the RANS/LES hybrid models,the detached-eddy simulation (DES)is efficient to simulate turbulent coherent structures.In the present work,the turbulent flow associated with a series of groynes was investigated using a zonal DES model(ZDES).The ZDES model augments the model's capability to overcome the model-stress depletion (MSD),which is a serious drawback of the RANS/LES hybrid models[11-12].The present work investigates the spatial turbulent coherent structures using ZDESfor different Froudenumbers.

1.Numerical model and simulation setup

1.1 Numerical model

The in-house codes(HydroFlow?)have been developed and calibrated in previous works[13-15].The numerical model usesσ-coordinate transformation in the vertical plane to fit the variation of the free surface and uneven bottom.The horizontal grid is unstructured,and the vertical grid is layered.The finite volume formulation of the Navier-Stokes equations is discretized using a second order total variation diminishing (TVD)scheme for the convective terms.A collocated predictor-corrector algorithm is used for the time integration,which is a semiimplicit scheme in time matching.In predictorcorrector methods for free surface water flow simulation[16-18],the fully hydrodynamic pressure is split into the hydrostatic and non-hydrostatic parts.The hydrostatic pressure is first calculated in the predictor step,and then the non-hydrostatic pressure is obtained in the subsequent corrector step by solving a discretized Poisson-typeequation.

Capturing the unsteady free surface is a critical issue in the numerical simulation of the free surface flow.A method named rigid lid approximation has been used to simulate open channel flows with groynes[8-10]and other structures[19-20].Although the rigid lid approximation is not as accurate in modelling unsteady free surfaces as some other methods,e.g.,the VOFmethod,it is generally considered acceptable for Froude numbers less than 0.5.The VOF method is powerful in modelling free surfaces,but the grid resolution near the interface between the air and water commonly needs to be refined,which leads to an increase in the computational cost.In the present 3-D model,the verticalσgrid is highly efficient and accurate in modelling the instantaneous free surface even when using a much coarser grid near it.Compared with the rigid lid approach,the coordinate transformation method is feasible even for higher Froude numbers.

1.2 Simulation setup

The sketch of the computational domain is shown in Fig.1,which is composed of a trapezoidal channel with a side slope of 1:3 slope as highlighted in Fig.2.The installed groynes are emerged and perpendicular to the main flow direction.The depth D=0.2 m in the main channel was chosen as the characteristic length scale,and the mean velocity U in the main channel was chosen as the velocity scale.The Reynolds numberDRe and the Froude numberDFr in Table 1 were calculated based on the variables U and D .In the computational domain,x is the stream-wise direction,and z is the vertical direction (originating at the still water level).In the span-wise direction,=0y corresponds to the location of the bank.The domain extends 20D upstream from the first groyne and 50D downstream from the last groyne.The thickness of the groyne is 0.25D.The width (3 )D over length (9 )D ( / )W L ratio of each embayment is 1/3.The depth in the embayment area increases from 0.15D at the sidewall to 1D with a constant slope of 1:3.The horizontal length of the slope bottom in the embayment is 2.5D,which is slightly shorter than the groyne length of 3D(Fig.2).Three sampling points were fixed in the third embayment to record the instantaneous velocity,which was labelled asa, b and c in Fig.3.

Fig.1 Schematic diagrams of thecomputational domain

Fig.2 (Color online)Computational mesh:(a)Mesh in the horizontal planecovering oneembayment,(b)Mesh around a singlespur dike,and (c)Mesh in a vertical crosssection

Table 1 Experimental deep water wave parameters

Fig.3 (Color online)Sketch showing the data recording positionsand the vertical crosssection S-S

A steady RANS type inflow boundary condition wasused.The flow at the inlet wasobtained by means of a pre-calculated fully-developed open channel flow with similar hydraulic parameters.At the outlet,a radiation boundary condition was used to limit the flow reflection.The coarser grid in the downstream RANSdomain plays a part in damping the small-scale flow disturbances generated within the upstream LES domain.

In the following study cases,the simulations were first run until the transient variations were eliminated.Statistical quantities were then calculated using the instantaneous flow fields over the next period of 120 /D U .The results were saved to hardware with 50 Hz in order to capture the high frequency turbulent flows.

2.Resultsand discussion

In the presentation and discussion of the results,the specific objectives included (1)analysing the instantaneous spatial and temporal characteristics of the vortex system,and (2)visualizing the major coherent structures presented by theaveraged flows.

2.1 Model validation

The present DES model has been successfully validated and used to simulate the free surface flow and flow past dunes[21].Although there were no measurements to validate the present numerical simulation,the numerical model was validated as being able to simulate free surface flow over the uneven bottom at approximate Froude or Reynolds number.The simulation accuracy can be guaranteed provided the grid resolution meets DESrequirements.The non-slip solid wall boundary condition was achieved by discretizing the boundary layer with fine grids,i.e.the first grid point to the solid wall being specified in the viscous sub-layer.Figure 4 presents thegrid resolution validation around the first spur dike tip.Figure 4(a)sketches the measuring positions for some critical parameters to determine the mesh quality.Figure 4(b)-4(d)show the wall units of the nearest grid points to the solid wall,in which the panel(b)presents the horizontal grid resolution adjacent to the dike wall,and panels (c)and (d) present the vertical grid resolution of the first grid to the bottom.Meanwhile,the interface positions between the RANS and LESzone are presented.The first grid points were all located in the viscous sub-layer,as required by the non-slip boundary condition.The interface of RANS/LES was located in the most vigorous turbulent activity in the boundary layer,which ensured a proper grid design for DES.

Fig.4 (Color online) Validation of the grid resolution on the solid wall,and the critical positions of the RANS/LES transition

2.2 Instantaneous flows

The main flow features include the formation of a horseshoe vortex (HV)system induced by the separated flow at the base of the groyne and the shedding vortex (SV)system induced by the separated flow at the groyne tip.The detached shear layers originate at the groyne tips near the free surface and develop downstream to form a multi-gyre recirculation in the embayment.

The vortical structures beneath the free surface were used to illustrate the distorted free surface.The instantaneousfree surface geometry is shown in Fig.5 for study cases C1 and C2.The free surface patterns,such as upwelling (“U ”in Fig.5),downdraft (“D”in Fig.5),and groove or ripple,were clearly observed.The simulations revealed that violent distortion of the free surface occurs near the tips of the groynes and at the interface of the embayment and main channel.The separated flow was powerful immediately downstream of the first groyne and gradually weakened downstream because of the shelter of the upstream groynes.A series of water whirlpools(“W ” in Fig.5)were advected downstream from the dike tip,which was the footprint of the SV.The shedding water whirlpool structures in case C1 were much clearer than those in case C2,which indicated that the intensive free surface distortion suppressed the SV.In case C2,large-scale grooves inclined across the interface of the embayment and the main channel,which plays an important role in the mass exchange process.The type of groove structure depends on the W /L ratio,the topography,and the FrDand ReDof the incident flow.It is well known that the instantaneous free surface geometry shows the “footprints” of the vortical structures just beneath the free surface.

The isosurfaces of (=6)Q for both cases are shown in Figs.6 and 7.The snapshots of the 1st(E1)and 3rdembayment (E3) were used to interpret the evolution of distinct vortices.The dominant coherent structures consisted of the necklace vortex wrapping the groyne tips just underneath the free surface,the shedding vortex beneath the free surface,and the horseshoe vortex around the base of the groynes.The shear layers created at the groyne tips started becoming unstable and then generated an SV.Near the free surface,the NV became stronger and suppressed the SV.The attenuation of the SV was not very significant in the study case C1,which has a lower Froude number.This is consistent with Vlachos and Tellionis’s[22]observation and Suh’s[23]simulation of flow past a free surface piercing a circular cylinder.Starting at the groyne tips,the NV developed downstream and distorted the free surface forming in ribbon-like vortex (RV).In deeper water,the NV was gradually diffused,and the SV was generated.Close to the rigid bottom,the SV was also suppressed,and a characteristic HV system was created.In front of the spur dike, the incoming bottom boundary layer flows separated and the vortices reorganized to generate the HV,which was clearly observed around the first groyne tip(Figs.6 and 7).The HV were stretched and elongated when the flow passed the groyne tips and were gradually torn and dissipated downstream of the domain.Due to the shelter of the upstream groynes,the HV became weaker when the flow wrapped the downstream groyne tips,which was clearly revealed by the comparison of the coherent structures in E1 and E3(Figs.6 and 7).

Fig.5 Magnified view of the instantaneous free surface,where U ,D and W represent upwelling, downdraft, and whirlpool,respectively

Fig.6 (Color online)Local turbulent coherent structures identified by isosurfacesof Q(flooded by vorticity magnitude)for C1

Fig.7 (Color online) Local turbulent coherent structures identified by isosurfaces of Q(flooded by vorticity magnitude)for C2

Fig.8 (Color online) Instantaneous out-of-plane vorticitiesat different water depths for C1.Panels(a),(b),and (c)in E1,and (d),(e),and (f)in E3

Figures 8 and 9 show contours of the out-ofplane vorticity Ωzat different water depths,where σ = -0 .05 represents the near-surface plane,σ = -0 .50 represents the mid-depth plane and σ =-0 .95 represents the near-bottom plane.Different from the classical seriesof shedding vorticesin opposite directions,only the vortices with negative vorticity Ωzwere clearly observed around the groyne tips and were convected downstream.The SV flow patterns were much clearer in the near-surface and the mid-depth planes.In the near-bottom plane,the vortical structure was affected by the HV,which elongated the SV stream-wise. The coherent structures were similar in E1 and E3,with vorticity inside E1 being a bit larger.In one embayment,besides patches of clockwise rotating vortices shed from the upstream dike tip into the embayment,patches of counterclockwise rotating vortices were reversely shed from the downstream dike tip into the embayment.The injection of flow from the downstream embayment into the upstream embayment was distinctly observed in E3,which played an important role in the water exchanging.The evolution of the shear layers is sketched using dotted lines in Figs.8 and 9.For both cases,the shear layers in E1 developed intensively downstream with a higher vorticity magnitude.In contrast,the shear layers were more fully-developed in E3 with a mild variation of the width and small magnitude of vorticity.

2.3 Averaged flows

The mean velocity fields were time-averaged over 120 /D U ,a time period long enough to ensure the convergence of the velocity statistics. Figure 10 shows an overview of the mean streamlines in the series of groyne.There was one recirculation at the corner in front of the first groyne dike,which was constrained by the sidewall and the slope bottom.In the subsequent embayments,a two-gyre circulation pattern was observed.A smaller circulation was created in the upstream corner of an embayment,and the larger one occupied the domain with most embayments.Some researchers have determined the effect of the embayment aspect ratio on the circulation pattern inside the embayment[24-25].In the present geometry condition, the two-gyre circulation pattern was reasonable and consistent with related research.The two-gyre flow pattern is much more clear in the first,third and fifth embayments.One possible reason is that the groyne field is not long enough to get a fully developed flow.Another possible reason may be the physical mechanism,i.e.,whether there is a periodic vortex motion when flow passing a series of groyne with a critical geometry.

Fig.9 (Color online)Instantaneousout-of-planevorticitiesat different water depths for C2.Panels(a),(b),and (c)in E1,and (d),(e),and (f)in E3

Fig.10 (Color online) Mean streamlines(flooded by streamwise velocity)in embayments for C2

Fig.11 (Color online) Local mean streamlines (flooded by streamwise velocity)in E3 for C2

Fig.12 Mean streamlines at different water depthsfor case C2

In order to clearly present the details of mean vortical structures inside the embayments,the local mean streamlines inside E3 are shown in Fig.11.Figure 11(a)clearly reveals the two-gyre circulation flow structure,in which one larger mean circulation spins upward to the free surface while the smaller one spins downward to the bottom.A section of the streamline was selected to present one possible trajectory of fluid particles into the embayment (see Fig.11(b)).The selected streamline started from a position at the mid-depth in the upstream embayment and passed the groyne tip into the target embayment.Along the streamline,the fluid particle moved gradually toward the free surface,and then turned into the inner area of the embayment resulting in a larger circulation.When approaching the upstream dike,the fluid particle turned around a reverse spiral axis,and went into the deep water,which resulted in a small steady circulation in the corner.Although the streamline presented in Fig.11(b)was elaborately picked up among millions of streamlines,it presented clearly one complex 3-D flow structure in the embayment with a sloped bottom.The mean flow pattern contributes to investigate the mass convection and diffusion in the embayment constrained by the solid obstructions and a critical slope bottom.

Fig.13 (Color online) Mean streamlines in different cross sections for C2,where the section site was labelled as S-S in Fig.3

Fig.14 (Color online)The mean turbulent kinetic energy at different water depthsfor C2

The panels in the middle column reveal the mean streamlines at mid-depth,i.e.,σ = -0 .50.A two-gyre pattern was the main coherent structure of the simulated turbulent flows in embayments.Compared with streamlines at the near free surface depth,the larger circulation structure was much clearer in embayments E1 to E5.In the vicinity of the downstream dike tip of one embayment,no flow was obviously injected from the next embayment into the present one.The effects of the free surface were distinctly suppressed at mid-depth.

The panels in the right column present the mean streamlines near the bottom,i.e.,σ = -0 .95.A distorted gyre flow pattern wascreated in embayments E1 to E5,where the smaller steady circulation disappeared.The cores of the flow circulations were pushed upstream.Around the downstream dike tip of the embayment,the flow was reversely injected from the next embayment.The centres of the circulation were nearly around the toe of the slope bottom.The presented steady flow structures through the water depth indicated different mass exchange passageways between themain channel and the embayments.

The details of the mean flow at vertical crosssections positioned at the middle of each embayment (labelled by S-S in Fig.3)were used to present the 3-D turbulent coherent structures.Figure 13 shows a two-gyre pattern in each section.The bigger gyre in the clockwise direction was formed around the slope toe,and the smaller one in the counter-clockwise direction was generated close to the side wall.The bigger clockwise circulation played an important role in the exchange of water and dissolved mass between the main channel and the embayments.The results revealed that the dissolved mass entered the embayment from the upper water body and left from the lower water body.In each selected cross-section,the opposing flows encountered near the slope bottom and went against the free surface.The two mean circulations force the masstransportation,for example the sediment.The characteristic flow structures contribute to the erosion and deposition of sediment.Sand bars are possibly formed at the flow stagnation points of the clockwise and counter-clockwise steady flow circulation.

Fig.15 (Color online) Frequency analysis of turbulent velocity at selected point a (C2)

The velocities at three sampling points were recorded and were examined using the spectral analysis.The velocity spectra were obtained using a Fourier transform with a Hamming window.Figures 15-17 shows the spectra of the stream-wise and transverse velocity.The spectrum of the stream-wise and transverse velocity displayed a –5/3 decay range indicative of the inertial subrange associated with the energy cascade from larger to smaller scales.Figure 17 showsa –3 decay rangein the velocity spectrum besides the –5/3 decay range,which is associated with a quasi-two-dimensional flow.The turbulence spectra highlighted 3-D characteristics of flows at the outer edge of the embayment,but indicated a 2-D flow structure inside the embayment, which is common in open channel flowswith groynes[10].

Fig.16 (Color online)Frequency analysis of turbulent velocity at selected point b(C2)

3.Conclusions

The hydrodynamics of flow in a straight trapezoidal open channel constructed with a series of groynes on the side slope was investigated using the ZDES.The instantaneous and averaged flows were investigated for cases with different incident flows.The simulations shed light on the coherent structures and the vortical dynamics around groynes,and the results are relevant to various environmental hydraulic issues.

Fig.17 (Color online)Frequency analysis of turbulent velocity at selected point c (C2)

The ZDES implementation used in the study accurately captured the vortical structures around groynes.The instantaneous turbulent coherent structures were explored by means of analysing the velocity and the second invariant of the velocity gradient tensor (isovalues of Q).The NV around the groyne tips just beneath the free surface,the SV over most of the water depth,and the HV generated at the groyne bases are the three primary types of vortical coherent structures.The spatial and temporal evolutionsof vortices are affected by the incident flow,topography and the arrangement of the groynes.The knowledge on the multi-scale vortices is important for investigating themassexchange.

The instantaneous turbulent flow reveals strong 3-D irregular flows around the tips and across the interface between the embayments and the main channel.The mean flows reveal several regular large-scale circulation structures including the gyre pattern insidethe embayment and the spatial evolution of vortical eddies.

Although the DES and LES have been successfully used for groyne flow simulations,the computational cost is still quite high because of the requirements on the spatial resolution.At present,it is challenging to develop advanced numerical models to simulate river flows using complicated real-life dimensions and geometries.