Numerical investigation of velocity distribution of turbulent flow through vertically double-layered vegetation

Nveed Anjum , Norio Tnk ,b,*

a Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan

b International Institute for Resilient Society, Saitama University, Saitama 338-8570, Japan

Abstract The velocity structures of flow through vertically double-layered vegetation (VDLV)as well as single-layered rigid vegetation (SLV)were investigated computationally with a three-dimensional (3D)Reynolds stress turbulence model, using the computational fluid dynamics (CFD)code FLUENT.The detailed velocity distribution was explored with a varying initial Froude number (Fr), with consideration of the steady subcritical flow conditions of an inland tsunami.In VDLV flows,the numerical model successfully captured the inflection point in the profiles of mean streamwise velocities in the mixing-layer region around the top of short submerged vegetation.An upward and downward movement of flow occurred at the positions located just behind the tall and short vegetation,respectively.Overall,higher streamwise velocities were observed in the upper vegetation layer due to high porosity,with Pr =98%(sparse vegetation,where Pr is the porosity),as compared to those in the lower vegetation layer, which had comparatively low porosity, with Pr = 91% (dense vegetation).A rising trend of velocities was found as the flow passed through the vegetation region, followed by a clear sawtooth distribution, as compared to the regions just upstream and downstream of vegetation where the flow was almost uniform.In VDLV flows,a rising trend in the flow resistance was observed with the increase in the initial Froude number,i.e.,Fr=0.67, 0.70,and 0.73.However, the flow resistance in the case of SLV was relatively very low.The numerical results also show the flow structures within the vicinity of short and tall vegetation, which are difficult to attain through experimental measurements.

Keywords:Vertically double-layered vegetation; Single-layered rigid vegetation; Numerical modeling; FLUENT; Velocity distribution; Turbulent flow

1.Introduction

Tsunamis cause damage to buildings,seaside artificial land,natural structures, and people's lives (Mori and Takahashi,2012).Disastrous tsunamis and storm surge events have demonstrated the need for protection measures in coastal regions to mitigate the effect of fluid force.Various researchers have performed numerical simulations to evaluate the effects of artificial/natural structures on reducing tsunamis (e.g.,Tanaka et al., 2014)or reducing/utilizing typhoon-induced storm surges along the coast (e.g., Zheng et al., 2017a,2017b; Wu et al., 2018).Coastal vegetation also plays a large role in tsunami mitigation(Shuto,1987).Coastal forests have attracted attention, as they play a role in trapping driftwood, dissipating the energy of flow, and providing soft landing places and escape routes (Tanaka et al., 2007).After the Indian Ocean Tsunami, various field observations and investigations have demonstrated the effectiveness of coastal vegetation on energy reduction of tsunamis (Kathiresan and Rajendran, 2005; Yanagisawa et al., 2009).In addition to the field observations,various experimental and numerical studies have been conducted to investigate the capability of coastal vegetation to mitigate the damage due to tsunami inundation.A numerical model was proposed by Harada and Imamura(2005)to evaluate the resistance created by pine trees through the drag and inertia forces.They showed that the force of water passing through the vegetation weakened, which minimized the damage behind the vegetation.Recently, an experimental study conducted by Pasha and Tanaka (2017)provided an understanding of the behavior of non-submerged vegetation with inland tsunami propagation.The study described the way in which a dense-vegetation forest reduced a greater amount of energy than a sparse-vegetation forest,but it is difficult to construct a dense forest in the actual field.

During the post-survey of the Indian Ocean Tsunami in 2004, in Kalutara, Sri Lanka, vertically double-layered vegetation (VDLV), comprised ofP.odoratissimusandC.equisetifolia, showed a strong ability to mitigate the disaster behind the vegetation (Tanaka et al., 2007).This opens the door to modeling VDLV in order to understand its effect and ability as a protection measure against tsunamis.To understand the basic flow and velocity distribution in an open channel, several numerical studies on vertically singlelayered vegetation (SLV)(Lima et al., 2015; Anjum et al.,2018a; Pu et al., 2019)have been performed.Some research has also been performed on VDLV(Liu et al.,2010;Huai et al.,2014;Anjum and Tanaka,2019)to investigate its influence on the velocity profile distribution in an open channel flow.The flow structures of open channel flows with sparsely and irregularly growing vegetation, which shows less resistance, are different from those with intentionally constructed coastal vegetation, which has a regular pattern and a dense arrangement in order to produce maximum resistance against the tsunami flow.No study about this type of VDLV as a countermeasure for the mitigation of tsunamis in inland areas has been reported yet.Hence, there is a need to explore the effect and importance of VDLV against tsunami flow, and to compare it with SLV.The protection of a coastal region using a natural defense system like vegetation is of great importance, as a coastal forest has the capability to reduce tsunami energy.Therefore, this study aimed to simulate the detailed velocity structure of inlandapproaching tsunami flows through coastal vegetation.

This study had a wide scope and a high utility in providing an understanding of the flow mechanism in the form of the detailed velocity structures around VDLV and SLV as well as their generated resistances.In addition, the velocity structure within the vegetation cylinders is difficult to capture in an experimental study because of the limitations that the velocity sensor of the electromagnetic flow meter is affected near the vegetation cylinders, and the laser light sheet in the Particle Image Velocimetry(PIV)cannot be used within the vegetation.Overcoming this difficulty was one of the advantages of the present numerical study.The study focused on numerical investigation of the steady subcritical inland tsunami flows.The objectives of this study were as follows:

(1)To analyze the detailed velocity distributions around both VDLV and SLV using a Reynolds stress model (RSM).

(2)To compare the mean flow characteristics against the varying initial Froude number in VDLV flows.

(3)To compare the effects of SLV and VDLV on the resulting flow structure.

2.Modeling approach

2.1.Experimental setup

The numerical model used for the present study was validated with the experimental data of Liu et al.(2010).Liu et al.(2010)performed the experiment in a 4.3 m-long and 0.3 m-wide flume with a constant slope of 0.003.A rigid VDLV,i.e.,a combination of tall vegetation(15.2 cm)and short vegetation(5.1 cm)with a diameter(d)of 0.635 cm,was installed in the channel,covering a section of 3 m in length and 0.3 m in width,bolted to the base at a distance of 1.3 m from the channel inlet.The short and tall vegetation cylinders were arranged in a staggered configuration with a spacing ofSs=5dandSt=10d(whereSsis the spacing between short cylinders andStis the spacing between tall cylinders), respectively.Instantaneous velocity measurements were taken 2.25 m downstream from the start of the vegetated section, ensuring a fully developed flow on average.A onedimensional laser Doppler velocimetry (LDV)system was used for the mean flow measurement.The rate of discharge and depth of flow were 1.14×10-2m3/s and 0.1212 m,respectively.Meanwhile, the Froude number and Reynolds number for the corresponding discharge were 0.282 and 37300,respectively.

2.2.Modeling setup and boundary conditions

The computational domain for the validation consisted of a periodic length of the vegetation arrangement in order to reduce the computational cost.Hence, a 0.6 m-long domain consisting of rigid VDLV was modeled, while all the other dimensions of the geometry were kept the same as those in the experiment of Liu et al.(2010)(Fig.1).The rigid vegetation was simulated with solid cylinders to achieve the flowstructure interaction.A tri-pave mesh scheme with tetrahedral elements was used in the simulation.The adopted mesh contained 300, 150, and 60 nodes in the streamwise, transverse, and vertical directions, respectively, which provided approximately 2.7 × 106grid cells.In order to certify the quality of computational fluid dynamics (CFD)simulations, a mesh independence trial was also performed.

A periodic boundary condition was used at the inlet/outlet in the streamwise direction,offering an interface between inlet and outlet faces (translational periodicity).A symmetry boundary condition was used at the free surface.A standard wall function and no-slip wall boundary conditions were used for treatment of the solid walls, i.e., the domain bed and cylinders.A mass flow rate of 1.14×10-2m3/s was provided at the periodic boundary.In combination with Reynoldsaveraged Navier Stokes (RANS)equations and a threedimensional (3D)RSM, simulations were conducted using the CFD code FLUENT,which uses the finite volume method for spatial discretization.The semi-implicit method for pressure-linked equations was used to achieve the pressurevelocity coupling.The standard values of under-relaxation factors were used in the simulation process.The convergence criteria of the residuals were set at 1×10-5.Details of the governing equations,turbulence model,and algorithms can be found in theFLUENT User's Guide.

Fig.1.Experimental setup.

2.3.Turbulence model validation

The numerical model results of streamwise velocities at specified positions were compared with the experimental results(Fig.2).Thex-axis shows the velocity,which is normalized by the shear velocity (u*= （gzS）1/2, wheregis the gravitational acceleration,zis the water depth, andSis the channel slope),whereas they-axis shows the depth of flow,which is normalized by the short vegetation height(hs).Both the experimental and numerical results show that the streamwise velocity magnitudes in the free-stream region(position 2)are comparatively higher than the velocities in the regions directly downstream and upstream of vegetation structures (positions 1 and 3).A strong agreement between the experimental and numerical data was observed, which demonstrates the validity of the present numerical model.However, a small difference between the experimental and numerical results is present close to the flow surface,i.e.,2.10 ＜z/hs＜2.22.This may be due to the symmetry boundary condition at the free surface, which treats the surface as flat.Also,there is a water level difference between the upstream and downstream of vegetation in reality due to the presence of emergent tall vegetation.This difference in water level causes the pressure difference,which can result in the increase of velocity close to the water surface (Roulund et al.,2005; Wang et al., 2014).Although small differences appear near the water surface, the internal velocity structures can be evaluated with respect to validation conditions, i.e., the symmetry boundary condition and short vegetation height.

Fig.2.Comparison of experimental and numerical results.

2.4.Simulation setup for present study

During the 2011 Great East Japan Tsunami, the tsunami flow was subcritical at many locations inundated by the tsunami,and the Froude number around an inland forest in the Sendai Plain in Miyagi Prefecture was between 0.7 and 1.0(Tanaka et al., 2013)with an approximate tsunami inundation depth between 7.3 m and 8.8 m.To set the flow conditions in this study, the Froude number and inundated flow depth similarities were applied to setting the model scale of the numerical experiment.The initial Froude number (Fr=U/（gz）1/2,whereUis the initial average velocity)was defined with the velocity and water depth, without consideration of a vegetation model.To set the subcritical flow conditions for an inundating tsunami, the water depths considered in the simulations were 6,8,and 10 cm,giving the initial Froude numbers of approximately 0.67, 0.70, and 0.73, respectively.

As the tsunami flow in an apparently steady subcritical state in the inland region was considered,the scale of the model was set at 1/100.The tsunami inundation and forest characteristics implemented in this study were not particular to any location but were considered generally, with the implication of similarities for some of the important parameters,e.g.,flow depth,Froude number, height of the tree trunk, and diameter of the tree trunk at the breast height in a real tsunami.The average trunk height of the tree (a pine tree)and diameter were found to be 15 m and 0.4 m, respectively, in Sendai City (Tanaka et al., 2012), and a tree could be modeled as a circular cylinder (Tanaka et al., 2014).Details about the vegetation configuration for simulation are given in Table 1.

The vegetation porosityPr(Iimura and Tanaka, 2012)is

wherentis the vegetation density.

In the present study,the vegetation models covered the full width and length ofWat the center of the computational domain, with the values of 30 cm and 24.27 cm, respectively,as shown in Fig.3.The vegetation density depends on the value ofG/d(Takemura and Tanaka, 2007).G/dvalues of 0.25, 1.09, and 2.13 represent dense, intermediate, and sparsevegetation, respectively (Pasha and Tanaka, 2017).Following the given definitions, two types of vegetation configurations,VDLV (cases 1, 2, and 3)and SLV (Case 4), were adopted(Table 1)with varying porosities and thickness of vegetation in a staggered arrangement.In cases 1, 2, and 3, a sparse arrangement of tall(emergent)vegetation was adopted,within which the intermediate short submerged vegetation was incorporated (Fig.3(a), (c)).In Case 4, only SLV of sparse arrangement was adopted for comparison purposes (Fig.3(b),(d)).Based on the porosity, the assembly of the short and tall vegetation models was divided into two different layers, i.e.,L1 and L2 in the vertical direction (Table 1), where L1 was within the level of short vegetation(5 cm)between the domain bed and top of short vegetation, and L2 was above L1,reaching the top of tall vegetation (15 cm)(Fig.3(c)).The lower vegetation layer had a high density of vegetation based on theG/dvalue of 0.563,and was considered a low-porosity vegetation layer,withPr=91%,whereas the upper vegetation layer had a relatively low density of vegetation (sparse arrangement)based on theG/dvalue of 2.125, and was considered a high-porosity vegetation layer, withPr= 98%,signifying the areas where the forest fringe may be damaged,and the soil around the trees may be scoured (Shuto, 1987).With a 1/100 model scale,the diameter of the circular cylinder was considered to be 0.4 cm,which was determined according to the observed inland Japanese pine trees (Tanaka and Onai,2017).The height of the short cylinder was set at 5 cm, in reference to the densely grown low-height Japanese coastal trees,whereas the height of the tall cylinder was set at 15 cm,keeping in view the average trunk height of the pine tree.

Table 1 Geometric conditions of vegetation configuration for simulation.

The computational domain dimensions used for the present study were similar to those used for validation, i.e., 0.6 m in length and 0.3 m in width.In the streamwise, transverse, and vertical directions,respectively,Case 1(as well as Case 4)had 30,150,and 30 nodes;Case 2 had 300,150,and 40 nodes;and Case 3 had 300, 150, and 50 nodes, which provided approximately 1.35 × 106, 1.8 × 106, and 2.25 × 106grid cells,respectively.All the boundary conditions were kept the same as those in the validation process.The hydraulic conditions for the four cases are summarized in Table 2.For the investigation of the velocity structure around SLV and VDLV, various important positions within the vicinity of vegetation cylinders were selected (Fig.3(a), (b)).Moreover, important horizontal surfaces and longitudinal sections (as shown in Fig.3(a), (b))throughout the computational domain were adopted for the detailed study of mean flow characteristics in the form of contour plot distribution.

Fig.3.Numerical modeling setup scheme(the extended part of the domain with red points shows the specified critical positions,the dashed red color lines show the specified longitudinal sections, and the units of dimensions are centimeters).

Table 2 Hydraulic conditions.

3.Results and discussion

3.1.Vertical distribution profiles of mean streamwise velocity

3.1.1.Velocity structure within short vegetation layer

The vertical profile distributions of the mean streamwise velocity (u)at the specified positions for all four cases are depicted in Fig.4(a)-(d).It can be observed that streamwise velocities were very low and decreased to their minimum close to the bed in all the cases.This effect of low velocities close to the bed region resulted from the resistance of the bed of the domain.Within the short submerged vegetation of VDLV,i.e.,z＜5 cm (Fig.4(a)-(c)), the magnitudes of streamwise velocities in the regions directly downstream of short and tall vegetation (positions 1, 5, 7, and 11)were small with large fluctuations, compared to those in the regions directly upstream of short and tall vegetation(positions 2,6,and 12)and free-stream regions (positions 3, 4, 8, 9, and 10).The reduction of the velocity behind the vegetation cylinders can be explained as the compensation of force acting on the cylinders due to flow over the cylindrical bodies.It can be clearly observed in all the cases that the velocity magnitudes within layer L1 were lower than those observed in layer L2.This effect was due to the fact that the porosity of the lower vegetation layer was relatively low, withPr= 91%, and the vegetation density was relatively high, withG/d= 0.563, as compared to those of the upper vegetation layer, which had a porosity and vegetation density equal toPr= 98% andG/d= 2.125, respectively.The velocities in the lower vegetation layer decreased by 40%-75% as compared to those in the upper vegetation layer.

3.1.2.Velocity structure around shear zone at top of short vegetation layer

Fig.4.Vertical distribution profiles of mean streamwise velocities at specified positions for different cases(the horizontal dashed line represents the top of the short submerged vegetation layer).

In VDLV flows, a sharp rise along with an inflection point in velocities at all the positions (except position 11)was observed close to the top of short submerged vegetation, i.e.,4.5 cm ＜z＜5.5 cm (Fig.4(a)-(c)).The model shows the ability to capture the inflection point in profiles of the mean streamwise velocity at the top of short vegetation.The sharp rise of the velocity close to the top of short vegetation resulted from the exchange of momentum between the top of the short submerged vegetation canopy and the overlying flow,where a shear zone existed.In almost all the profiles, a gradient of velocity occurred over the mixing-layer zone.This mixinglayer zone is a region of maximum vorticity with a large number of vortices(Singh et al.,2019).The profiles remained almost undeflected beneath the mixing-layer zone, i.e., below the height of short vegetation (z＜4.5 cm).This inflection point in the velocity profile close to the top of the short submerged vegetation is similar to that observed in previous studies for SLV flows (e.g., Lopez and Garcia, 2001; Righetti and Armanini, 2002)and VDLV flows (Anjum et al., 2018b;Rashedunnabi and Tanaka, 2018; Ghani et al., 2019).In contrast, the vertical profiles of the streamwise velocity at position 11 showed no sharp inflection even at the top of short vegetation in all the VDLV cases.This was because this position was located directly downstream of tall emergent vegetation, where the vegetation continued to generate resistance to the flow,resulting in low velocities followed by large fluctuations.The drag force affected the profile since the tall cylinders were not completely submerged.

3.1.3.Velocity structure above top of short vegetation

Fig.4(a)-(c)shows that the streamwise velocities were relatively higher above the top of the short vegetation layer at almost all the specified positions.This was because the resistance of vegetation in the upper vegetation layer was reduced due to the lower vegetation density, and the flow had relatively higher velocities in this region.Moreover, the velocities at positions 2,11,and 12 above short vegetation were lower than the velocities at other positions located in freestream regions.This was because these positions were located in the regions directly upstream and downstream of tall vegetation, and in line with tall emergent vegetation, respectively.Also, due to the downstream affected region of tall emergent vegetation, the velocities at positions 9, 10, and 12 decreased in the vicinity of the flow surface.

3.1.4.Effect of initial Froude number and vegetationconfiguration on vertical distribution profiles of streamwise velocity

With the increase of the initial Froude number, while the VDLV configuration remained constant (Fig.4(a)-(c)), the streamwise velocity magnitudes at almost all the specified positions increased slightly, resulting in greater sharpness in the velocity profiles just at the top of short submerged vegetation.The vertical flow structure of streamwise velocities for SLV differed from that for VDLV.In the SLV case(Fig.4(d)),the vertical profiles of streamwise velocities at all the specified positions were constant along the depth of flow, except at the position directly downstream of tall vegetation (position 9).The velocity magnitude at position 9 was lower than that at the other positions, while its velocity fluctuation was relatively large.However, an insignificant influence of tall emergent vegetation was observed at positions 7,8, and 10, resulting in slightly lower magnitudes of velocity compared to those at other positions located in the free-stream regions.

3.2.Vertical distribution profiles of mean transverse velocity

Fig.5 shows the vertical profile distributions of the mean transverse velocity (v)at the specified positions for all the cases.As compared to the streamwise velocities, the transverse velocities at all the specified positions for VDLV (cases 1, 2, and 3)were very low, with slight fluctuations in the profiles.At almost all the positions, the magnitudes of transverse velocities were close to zero.At positions 3 and 10(located in lower free-stream regions between the lines of short submerged vegetation in a diagonal pattern), a slightly positive transverse movement of flow occurred along the vertical depth of flow up to the height of the lower vegetation layer, and then the transverse velocities became slightly negative or zero abovez= 5 cm due to the increase in porosity and decrease in resistance only caused by tall emergent vegetation.Moreover, the transverse flow deflection in the positive and negative directions was more prominent at positions 8 and 4 (located in free-stream regions between the lines of short submerged and tall emergent vegetation),respectively.This was due to the influence of the tall emergent vegetation structure,which generated resistance to the flow up to the surface, and resulted in the flow being deflected more significantly in the opposite directions (positive and negativey-axis)to its location.It can be observed that the fluctuation in the profiles of transverse velocities increased with the initial Froude number (Fig.5(a)-(c)).The transverse velocity magnitudes as well as fluctuations in the vertical profiles for SLV in Case 4 (Fig.5(d))were much less than those for VDLV in Case 1 (Fig.5(a)).However, due to the influence of tall emergent vegetation,low negative values of the transverse velocity occurred at positions 4, 6, 7, and 8.

Fig.5.Vertical distribution profiles of mean transverse velocities at specified positions for different cases.

3.3.Vertical distribution profiles of mean vertical velocity

The vertical profile distributions of the mean vertical velocity(w)for all the cases are illustrated in Fig.6.The positive vertical velocity shows upward movement, whereas the negative vertical velocity shows downward movement of flow.Similarly to the transverse velocity, the vertical velocity showed magnitudes close to zero, with slight fluctuations in the profiles along the depth of flow at almost all the positions in both SLV and VDLV cases.Generally, very low vertical velocities were observed at all the specified positions in VDLV cases (Fig.6(a)-(c)).However, a vertical movement of flow occurred just at the top of short submerged vegetation.Within the region of the top of short submerged vegetation, i.e.,4.5 cm ＜z＜5.5 cm,a negative flow deflection occurred at the positions located downstream of short submerged vegetation as well as upstream of tall vegetation(positions 1,2,3,4,5,7,and 8),whereas the flow was deflected in the positive direction at the positions located downstream of tall emergent vegetation (positions 9, 11, and 12).The increase of the initial Froude number (Fig.6(a)-(c))resulted in a slight increase in fluctuations of vertical velocity profiles, especially for the positions located directly upstream and downstream of the short and tall vegetation structures (positions 1, 2, 5, 6, 7, 11,and 12).Regarding the case of SLV (Fig.6(d)), the values of vertical velocity were close to zero, and the profiles were almost constant along the depth of flow.Hence,the upward or downward movement of flow in the SLV case was almost negligible.

3.4.Contour plot distributions of streamwise velocity

3.4.1.Velocity structure along horizontal surfaces

The simulated contour plot distributions of the mean streamwise velocity along the free surface and top surface of short submerged vegetation atz=5 cm for Case 1 and Case 4 are presented in Figs.7(a), (b)and 8(a), (b), respectively.It can be observed that the velocity decreased to its minimum just behind the vegetation structures, whereas it increased to its maximum in the adjacent regions in theydirection.In the VDLV case (Fig.7), the flow velocity decreased by approximately 45% directly downstream of vegetation, whereas it decreased by approximately 40% in the SLV case (Fig.8), as compared to that in the regions directly upstream of vegetation, showing that the flow was largely obstructed by the vegetation structures.The existence of vegetation had a noticeable effect on the velocity distribution, resulting in a large amount of turbulence in the form of variation of velocities within the vegetation region as well as in the downstream region.Wake regions in the form of primitive K′arm′an vortex(PKV)streets behind the individual vegetation structures in L2, with aG/dvalue of 2.125 (Figs.7(a)and 8(a), (b)),showed consistency with the results of previous research by Takemura and Tanaka (2007), in which they observed a PKV street behind the individual cylinder in the case ofG/d＞1.8.The velocities at the top surface of L1 in the VDLV case were lower(Fig.7(b))than those at the free surface.This was due to the larger resistance of both short and tall vegetation structures at the top of the short vegetation layer,at which the porosity of vegetation was relatively low, withPr= 91%, whereas, only tall vegetation structures played a role in reducing the flow at the free surface, at which the porosity of vegetation was comparatively high, withPr= 98%.However, no clear difference in velocity values was observed between the surface atz=5 cm and the free surface for SLV(Fig.8(a),(b)).This was because of the constant porosity(Pr=98%)of the vegetation layer.

3.4.2.Velocity structure along longitudinal sections

Fig.6.Vertical distribution profiles of mean vertical velocities at specified positions for different cases.

Fig.7.Contour plot distributions of mean streamwise velocities along horizontal surfaces and longitudinal sections in Case 1.

The contour plot distributions of the streamwise velocity through the longitudinal sectionsy=15 cm andy=14.375 cm in the VDLV case (Fig.7(c), (d))also show that the flow was largely affected by the presence of vegetation.An increase in the velocity relative to the initial flow velocity above the short submerged vegetation layer was observed along the entire computational domain, and was more clearly visible in the vegetation region.This governed the transfer of mass and momentum between the short submerged vegetation canopy and the overlying flow when vortices appeared due to shear layer formation at the inflection point atz= 5 cm (Finnigan, 2000;Ghisalberti and Nepf, 2005).In the numerical study of flow through VDLV,Barrios-Pi~na et al.(2014)also found an increase of velocities above the region of submerged vegetation.A clear difference in velocity distributions between the regions above and within the short vegetation layer can be observed,showing consistency with those observed in Fig.4(a)-(c).It can also be observed from the velocity contour through the sectiony= 15 cm (Fig.7(c))that the velocity was comparatively low directly behind the tall vegetation structures and gradually rose towards the top of the short vegetation structures downstream.Hence,the deceleration effect and resistance resulting from tall vegetation structures were more prominent than those of short vegetation structures.This effect is difficult to capture in an experimental study.Meanwhile, the distribution of velocity through the sectiony= 14.375 cm (Fig.7(d))shows that the flow was only obstructed by short vegetation structures within the short vegetation layer, and the velocities continued to increase above the submerged vegetation due to the exchange of momentum between the vegetation top and the overlying flow,resulting in a growing pattern of velocities above layer L1.Moreover,the flow passing through SLV(Fig.8(c),(d))was not greatly affected.The velocities upstream and downstream of the vegetation region remained almost constant, whereas the obstruction could mainly be observed within the vegetation region where the velocity magnitudes were reduced to a minimum just behind the tall emergent vegetation (Fig.8(c)).An increase in velocity could also be seen through the sectiony= 14.375 cm (Fig.8(d))as the section passed between the center line(adjacent regions)of tall vegetation structures where higher velocities were found relative to the initial velocity.

Fig.8.Contour plot distributions of mean streamwise velocities along horizontal surfaces and longitudinal sections in Case 4.

3.5.Distributions of streamwise and transverse velocities along domain length

3.5.1.Velocity structure along domain length

The streamwise velocity and transverse velocity profile distributions along the domain length at the sectiony= 15.3125 cm for the VDLV cases, and at the sectiony= 15.625 cm for the SLV case are presented in Figs.9 and 10, where the dashed box represents the vegetation region.In all the cases,the flows through the vegetation region appeared to be non-uniform, whereas they were almost uniform in the upstream and downstream regions.The flow velocities were observed to be significantly higher in the vegetation region than those observed in the upstream and downstream regions,exhibiting a sawtooth distribution of velocity magnitudes in the vegetation region.In Case 1, the streamwise velocity atz=5.5 cm in the vegetation region gradually increased along the longitudinal direction (Fig.9(a))due to a very low submergence level of the short vegetation layer, which slightly influenced the velocity structure in the upper vegetation layer.However, the increase in the streamwise velocity in other VDLV cases(Fig.9(b),(c))was not observed in the vegetation region as the submergence level of the short vegetation layer was increased.This shows that the relative submergence level of the short submerged vegetation layer influences the velocity structure in the upper vegetation layer.With the presence of vegetation (Pr= 91% and 98% for layers L1 and L2,respectively), the flow passing through the vegetation region showed a noteworthy difference from that in the upstream and downstream regions without vegetation (Pr= 100%).The rising streamwise velocities in the vegetation region were rational due to the resistance and obstruction of the vegetation structures.The velocities in the vegetation region increased by 11%-15% and 8%-12% for VDLV flows and SLV flows,respectively, as compared to those in the upstream and downstream regions without vegetation.

Moreover, the sawtooth distribution of streamwise velocities prominent in the vegetation region (Figs.9 and 10)is difficult to observe within the vegetation structures in the experimental investigation.This demonstrates another advantage of this numerical modeling.Within the upper vegetation layer atz= 5.5 cm, an abrupt rise and fall in velocities occurred, resulting in a significant distribution of the velocity.This phenomenon was due to staggered arrangements of short submerged vegetation and tall emergent vegetation.The streamwise velocity initially rose normally as the flow passed through the overlying region of a short submerged vegetation structure, fell normally as the flow passed through the upstream vicinity region of a tall emergent vegetation structure, then showed a rapid rise as the flow reached the adjacent region of the emergent vegetation structure, and finally showed a rapid fall (even low values compared to the velocity values observed upstream and downstream of the vegetation region)when the flow reached the region downstream of the tall emergent vegetation structure, and this distribution continued throughout the vegetation region.However, the sawtooth distribution in the lower vegetation layer atz=4 cm was almost constant.The rise and fall in streamwise velocities occurred as the flow reached the adjacent regions and the regions downstream and upstream of diagonally arranged staggered short and tall vegetation structures, respectively.This phenomenon can be easily understood in Figs.7 and 8.Furthermore,the transverse velocity also showed similar trends in all the cases.Within the vegetation region, the maximum transverse velocity occurred where the streamwise velocity was at its minimum, i.e., the lateral movement of flow occurred when the longitudinal movement of flow was lessened due to the resistance of vegetation.However, the magnitudes of transverse velocities were very low compared to those of streamwise velocities,and were observed to be negligible just upstream and downstream of the vegetation region.

Fig.9.Variations of stream wise velocities along domain length for different cases.

Fig.10.Variations of transverse velocities along domain length for different cases.

3.5.2.Effect of initial froude number and vegetation configuration on velocity structures

Due to the high porosity in the tall vegetation layer, the velocity magnitudes were higher in the longitudinal section above the submerged vegetation layer atz=5.5 cm,followed by a small number of rises and falls due to the presence of a sparse vegetation structure (only with tall vegetation).In contrast, the velocities in the longitudinal section below the submerged vegetation layer atz= 4 cm experienced a large number of rises and falls due to the presence of a dense vegetation structure(with both tall and short vegetation).With the increase in the initial Froude number, a slight increase in velocity magnitudes was observed in layer L2 atz= 5.5 cm,whereas this effect was not clearly visible in layer L1 atz= 4 cm (Fig.9(a)-(c)).The difference in velocity distributions along the domain length atz= 4 cm andz= 5.5 cm for SLV(Fig.9(d))was almost negligible,due to the presence of only a tall vegetation layer, with a constant porosity(Pr=98%).Moreover,the amplitudes of rising and falling of velocities in SLV flows were less than those observed in VDLV flows,indicating that the flow instability and turbulence caused by VDLV were significantly higher than those caused by SLV.

4.Conclusions

The flow responses approaching VDLV and SLV under steady subcritical flow conditions (with the initial Froude numbers of 0.67, 0.70, and 0.73)were investigated computationally using a CFD code FLUENT.The following conclusions were drawn:

(1)In VDLV flows, almost all the vertical profiles of the mean streamwise velocity showed a sharp inflection point at the top of the short submerged vegetation layer,with sharpness increasing slightly with the increase of the initial Froude number.The inflection was due to the vertical exchange of momentum between the flow at the top of short submerged vegetation and the overlying flow.The interaction of slow and fast flows over different layers near the top of short vegetation generated a velocity gradient, and resulted in a significant mixing layer in this region.

(2)Within the short submerged vegetation layer, the mean streamwise velocity significantly decreased by 40%-75%due to thelowporosity(Pr=91%),withadensevegetationarrangement ofG/d=0.563,ascompared tothatin the upper vegetation layer,which had a relatively high porosity of vegetation(Pr=98%),with a sparse vegetation arrangement ofG/d=2.125.

(3)In the regions directly behind the vegetation structures,the flow velocity decreased by approximately 45% for the VDLV flow, whereas it decreased by approximately 40% for the SLV flow.

(4)The streamwise velocities rose by 11%-15% in the vegetation region for the VDLV flow,whereas they rose by 8%-12%for the SLV flow,as compared to the upstream and downstream regions without vegetation.The flow velocities followed sawtooth distributions in the longitudinal section within the vegetation region, where the velocity fluctuation in the upper vegetationlayerslightlyincreasedwiththeinitialFroudenumber.

(5)The flow structure through SLV differed from that through VDLV.Almost constant profiles of mean flow characteristics in the SLV flow appeared with small fluctuations compared to those in the VDLV flow, which shows that SLV may not be very effective in producing resistance to the tsunami flow.

This study revealed the effectiveness of the combination of short and tall vegetation in flow resistance.The addition of a short submerged vegetation layer (G/d= 0.563)within an emergent vegetation arrangement(G/d=2.125)resulted in an increase in flow resistance.Hence,this arrangement of VDLV can be used as an effective measure for tsunami mitigation,which is easy to construct in the field as compared to dense emergent vegetation.Furthermore, the inflection point in VDLV flows observed at the top of short submerged vegetation in this study showed that the flow had the maximum effect of momentum exchange in this region,and there was a noticeable velocity gradient effect here.The potential of the two-layer vegetation arrangement to provide resistence against a tsunami flow needs further investigation based on twodimensional (2D)depth-averaged flow modeling.In addition,further 3D study on the turbulence flow characteristics is also needed to clarify our fundamental understanding of the effectiveness of VDLV over SLV.This study can be used to enhance our knowledge of vegetation designs for future planning of flood protection and vegetation-based defense systems for tsunami mitigation.