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Ocean Engineering 33 (2006) 1896–1918

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Numerical study on the flow characteristics oftidal jets induced by a tidal-jet-generator

J.-C. Parka,�, T. Okadab, K. Furukawab, K. Nakayamab

aDepartment of Naval Architecture and Ocean Engineering, Pusan National University, 30,

Jangjeon-dong, Gumjeong-gu, Busan, Republic of KoreabMarine Environmental Division, Port and Airport Research Institute, 3-1-1 Nagase, Yokosuka, Japan

Received 30 June 2005; accepted 19 October 2005

Available online 15 March 2006

Abstract

The flow characteristics of tidal jets induced by a Tidal-Jet Generator (TJG) are investigated

using a finite-difference numerical scheme, named Navier–Stokes (NS)–Marker and Cell

(MAC)-TIDE, based on the fully 3D NS equations. The TJG is an enclosed rectangular

breakwater, which has vertical opening and a large enclosed volume inside. During both

phases of tide, strong and uni-directional jets can be obtained locally from the inlet of the

TJG, due to the water level difference between the inner and outer sides of TJG.

The computed results are extensively compared with three other independently developed

numerical models; 3D-ADI, DVM, and CIP-CSF. These models are based on quasi-3D, 2D

depth-averaged, and fully 3D NS equations, respectively. It is seen that the present fully 3D

numerical model NS–MAC-TIDE can predict the maximum intensity of inlet velocity with

higher accuracy than the other numerical models when compared with the empirical function

proposed from the experiments. The numerical simulations based on NS–MAC-TIDE can

reproduce successfully the processes of generation, development, and dissipation of tidal jets.

The effects of gap opening on the main characteristics of the tidal jet flow are assessed.

Through numerical assessment, it is also clearly demonstrated that the residual time of a

pollutant distributed around the front of the TJG can be decreased by significant amount due

see front matter r 2006 Elsevier Ltd. All rights reserved.

.oceaneng.2005.10.024

nding author. Tel.: +8251 510 2480; fax: +82 51 581 3718.

dress: jcpark@pusan.ac.kr (J.-C. Park).

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J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–1918 1897

to the locally induced tidal jet. The TJG can thus utilize tidal energy for water purification in

local marine environment by providing a flushing mechanism.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Tidal-jet generator (TJG); Tidal jet; Fully 3D Navier–Stokes equation; Finite-difference

method; Wave-height function; Gap flow; Opening ratio; Residual flow; Sub-grid scale (SGS) turbulence

model; Water purification

1. Introduction

In the vicinity of tidal inlets, such as estuary, river/bay mouths, and portentrances, tide-induced currents play a significant role in various coastal/environ-mental processes. Furthermore, in the presence of a narrow entrance, currents canproduce strong and uni-directional jet flows in the neighborhood of such a suddenconstriction. The jet is known as a tidal jet when it is solely caused by the tidalforcing. During both phases of tide, the flow issuing from the inlet forms a turbulentjet and generates eddy pairs of high vorticity near the jet-head. The large eddy vortexpair created by the jet entrains the surrounding waters; consequently affecting thewater mass distribution and transport processes in the jet-influenced domain. As theflow develops, momentum dissipation in the jet grows gradually due to the influenceof vorticity near the jet-head. These hydro-mechanical processes are repeatedpersistently on both sides of the inlet with the period of tide. The processesconsequently affect sediment and mass transport.

A number of authors (French, 1960; Taylor and Dean, 1974; Ozsoy, 1977;Unluata and Ozsoy, 1977; Joshi, 1982; Joshi and Taylor, 1983; Ismail and Wiegel,1983; Wolanski et al., 1988) have investigated the general solutions for the jet widthand the centerline velocity variation of fully developed jets under the assumption ofself-similarity and quasi-steady turbulent jet behavior. However, these solutions donot cover one case of fundamental interest, the evaluation of the maximum intensityof a starting jet before development occurs. This case is of interest because it willdirectly influence the flow patterns/structures in the area neighboring a narrowentrance. If a bay has a very thin vertical opening, tidal forcing may give rise to largedifference in surface elevation between the two sides of the entrance. Then,dissipation should be enhanced by strong water jets falling down through the inlet,and the corresponding 3D effects, such as upwelling motion in the near field of inlet,may play an important role.

In order to utilize the effect of tidal jet and control coastal environment in astagnated area, where resolved nutrients and/or littoral effluent materials concentrate,a Tidal-Jet Generator (TJG) was proposed (Furukawa et al., 1994). The TJG is anenclosed rectangular breakwater/reservoir, which has a vertical opening and largeenclosed volume inside. During both phases of tide, strong and uni-directional jets aregenerated near the inlet of the TJG due to the water level difference between the innerand outer sides of reservoir. The residual time of a pollutant may be decreased by thisflow. It means that the TJG can be used as a flushing system for water purification.

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J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181898

In the present study, the characteristics of flow field around a TJG are investigatednumerically using a finite-differencing scheme, named Navier–Stokes (NS)–Marker and Cell (MAC)-TIDE method, which is based on the fully 3D NSequations and a modified MAC algorithm. The numerical method is brieflyintroduced in Secion 2.

The numerically predicted maximum intensity of the jets is extensively comparedwith the hydraulic experiments of Furukawa et al. (1994) and three otherindependently developed numerical models, namely, 3D-ADI method (Leendertse,1989; Furukawa and Hosokawa, 1994), Discrete Vortex Method (DVM) (Furukawaand Wolanski, 1998), and CIP-CSF method (Nakayama and Satoh, 1999). The briefdescription of the three numerical models is given in Secion 3. In Section 4, thenumerical convergence tests, the results of comparative study, and the flowcharacteristics of TJG-induced tidal jets are discussed. The possibility of local waterpurification by the TJG is also assessed numerically. Some concluding remarks aregiven in Section 5.

2. Finite difference method, NS–MAC-TIDE

2.1. Governing equation

Assuming that the fluid is incompressible and homogeneous, the governingequations are given by the following NS equation and continuity equation:

quqt¼ �r

p

rþ

2

3k

� �þ a, (1)

r � u ¼ 0, (2)

where

a ¼ �ðu � r Þuþ ðnþ nSÞr 2uþ ðr nSÞS þ f, (3)

S ¼

2 quqx

quqyþ qv

qxquqzþ qw

qx

qvqxþ qu

qy2 qv

qyqvqzþ qw

qy

qwqxþ qu

qzqwqyþ qv

qz2 qw

qz

26664

37775. (4)

In the above equations, u ¼ ðu; v;wÞ is the velocity vector, p the pressure, t thetime, r the gradient operator, n the kinematic viscosity for laminar flow, nS the eddyviscosity coefficient of the subgrid-scale (SGS) turbulence model, and f is the externalforce including the gravitational acceleration, g( ¼ 9.8m/s2). The density r isassumed constant over the fluid region. The SGS turbulence model is introduced inthe same manner as in Miyata and Yamada (1992). According to the Smagorinsky’sassumption (1963), the SGS eddy viscosity nS and the turbulent kinetic energy k are

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J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–1918 1899

derived as follows:

nS ¼ ðC1DÞ2 qui

qxj

qui

qxj

þquj

qxi

� �� �1=2, (5)

k ¼nSð Þ

2

C0Dð Þ2, (6)

where C1 ¼ 0:1 and C0 ¼ 0:094 for the present study, and D is the length scale set atD ¼ ðDxDyDzÞ1=3.

In the time-marching procedure, the solutions of the governing equations in eachregion are obtained separately at each time step. At the free surface, the followingkinematic and dynamic conditions can be applied neglecting viscous stress andsurface tension:

DðZ� zÞ

Dt¼ 0, (7)

p ¼ pair. (8)

Here, D is the total derivative, Z denotes the free-surface profile, so-called thewave-height function, and z represents the vertical coordinate. Eq. (7) means that theparticle on the free-surface moves with the free-surface. After determining the free-surface location, the governing Eq. (1) is integrated in the computational domain.

2.2. Algorithm and differencing scheme

The solution algorithm is similar in substance to the NS–MAC-NWT (Park et al.,1999) and the TUMMAC method (Miyata and Park, 1995) in which the velocity andpressure points are defined in a staggered manner in the framework of a rectangularcoordinate system. In the time-marching process, the distribution of wave-height iscalculated from Eq. (7) and the new pressure field is determined by iteratively solvingthe following Poisson equation:

r � rP ¼ r � aþuðnÞ

Dt

� �¼ r � b, (9)

Pmþ1 ¼ Pm þ o0ðr � rP� r � bÞ, (10)

where P ¼ p=r and o0 is the relaxation factor set at a value smaller than unity. Thesuperscript n denotes the time level and m the iteration level. The b term includesboth diffusion and convection terms and is called the source term. The velocity fieldis updated through the NS Eq. (1) and the boundary values of the velocities are thenset at the new location of the interface. Eq. (1) is solved by time-marching and atevery time step, and Eq. (9) is solved by an iterative procedure. The computation isrepeated until the number of time steps reaches the predetermined final value.

The finite-difference scheme for the convective terms must be carefully chosen,since it often renders decisive influences on the results. In the present simulation, for

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J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181900

the convective term a flux-split method such like a third-order upwind scheme withvariable mesh size is employed in space so that variable mesh system can be used forall three directions. The second-order Adams–Bashforth method is used for thetime-differencing. For the diffusive terms a second-order central differencing schemeis employed.

The transport Eq. (5) of wave-height function is approximated by a fully implicitmethod:

apZnþ1ij ¼ aeZnþ1

iþ1j þ awZnþ1i�1j þ anZnþ1

ijþ1 þ asZnþ1ij�1 þ c, (11)

where

ae ¼ �Dtðup � jupjÞ

2Dxiþ1=2, (12a)

aw ¼Dtðup þ jupjÞ

2Dxi�1=2, (12b)

an ¼ �Dtðvp � jvpjÞ

2Dyjþ1=2

, (12c)

as ¼Dtðvp þ jvpjÞ

2Dyj�1=2

, (12d)

ap ¼ 1þ ae þ aw þ an þ asð Þ, (12e)

c ¼ Znij þ wpDt. (12f)

Here, the subscript (i, j) indicates the discretized location of wave-height. Thesolution of discretization Eq. (11) can be obtained by a standard Gaussian-elimination.

The dynamic free-surface condition of Eq. (8) is implemented by the so-called‘‘irregular star’’ technique (Chan and Street, 1970) in the solution process of thePoisson equation for the pressure. The pressure at the free-surface is determined byextrapolating with zero gradient in a direction approximately normal to the freesurface and taking into consideration the static pressure difference in the verticaldirection due to gravity. Similarly, velocities are extrapolated at the interface withapproximately no normal gradient. This treatment is in gross accord with the viscoustangential condition at the free surface.

2.3. Other boundary conditions

The no-slip boundary condition is imposed on enclosed solid walls including thebreakwater. All velocity components are set to zero on the boundary, while thepressure and the values of wave height are extrapolated by applying the no-normal-gradient condition in horizontal direction. In the vertical direction, however,pressure is extrapolated by considering the static pressure differences.

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A forcing tide at the open boundary is given by

Z ¼ A cos2pT

t

� �¼ A cos st, (13)

where A is the amplitude, T the period of tide, s the angular frequency of tide, and t

the time. The forcing tide will be started from a maximum ebb tide. The staticpressure is set at the open boundary according to the time-movement of sea levelgiven in Eq. (13).

3. Other numerical models

The computational results from the NS–MAC-TIDE are compared with threeindependently developed numerical models; 3D-ADI, DVM, and CIP-CSF. Themain features of the numerical models are described briefly in this section and arealso summarized in Table 1.

3.1. 3D-ADI method

This finite-difference model developed for the computation of 3D free-surfaceflows solves the continuity and NS equations assuming the hydrostatic pressure inthe vertical direction. With this assumption, the momentum equations to be solved

Table 1

Characteristics of various numerical models

NS–MAC-TIDE 3D-ADI DVM CIP-CSF

Discretization FDM FDM DVM FDM

Governing Eq. Fully 3D NS 3D NS using

hydrostatic

assumption

2D depth-averaged

NS

Fully 3D NS

Grid system Rectangular Rectangular — Curvilinear

Calculation point Staggered mesh Staggered mesh Released vortex points

(Lagrangian)

Staggered mesh

Time integration Explicit (second-

order

Adams–Bashforth) ADI

Explicit (Euler) Explicit (Euler)

Differencing of

convective term

Third-order

upwind

Third-order

upwind

Lagrangian CIP

Eddy viscosity By SGS model Constant (0-

equation)

Implicit (vortex core) By SGS model

Boundary

condition (BC) of

free-surface

(Kinematic)

Wave-height

function Eq.

Depth-averaged

continuity Eq.

Depth-averaged

continuity Eq.

ALE

BC on vertical wall No-slip No-slip Slip No-slip

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J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181902

become:

qfqtþ u

qfqxþ v

qfqyþ w

qfqz

¼ �1

rrPþ nh

q2fqx2þ

q2fqy2

� �þ nz

q2fqz2

� �, ð14Þ

qP

qz¼ �rg, (15)

where nh and nz are the horizontal and vertical eddy diffusivity, respectively. InEq. (14) above, the dependent variable f represents repeatedly the horizontalvelocity components, u and v.

A 3D Alternated Directions Implicit (ADI) algorithm (Leendertse, 1989;Furukawa and Hosokawa, 1994) is introduced in the time-marching procedure,which allows the calculation of a 3D system with large time steps to be stable and notto be constrained by surface wave propagation.

3.2. Discrete vortex method

The DVM is based on the 2D depth-averaged NS equation and the conservativeequation. The model has been adapted for use under the boundary element method(BEM) to account for solid boundaries (Kuwahara, 1978).

In a coordinate system moving with the circulation G �R

SodS, where o is the

vorticity and S the area disturbed by the vorticity, the governing equations aresimplified to

dGdt¼ �gb

juj

hGþ KXr

2G, (16)

where h the water depth, KX the horizontal eddy viscosity coefficient, and gb thebottom friction coefficient initially taken to be equivalent to Manning’s frictioncoefficient.

The left-hand side of Eq. (16) represents the rate of change of the circulation alongthe path of the discrete vortex, while the first and second terms on right-hand siderepresent the effect of bottom friction and the diffusion of vorticity, respectively. Thedevelopment of the circulation due to an individual vortex of circulation G iscalculated using a Lagrangian method. A vortex is assumed to have a core which hasGaussian distribution of vorticity:

o ¼G

pd2exp �

r2

d2

� �, (17)

where r is the distance from the center and d is the diameter of the vortex core whereinduced velocity becomes maximum. As a result of diffusion, d is assumed toincrease with time as follows:

d ¼ffiffiffiffiffiffiffiffiffiffiffi4KX t

p. (18)

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The tangential velocity ur induced by this vortex is

ur ¼G2pr

1� exp �r2

4KX t

� �� �. (19)

The details of solution algorithm are described in Furukawa and Wolanski (1998).

3.3. CIP-CSF method

The CIP-CSF method has been developed for large-eddy simulations (LES) ofgravity currents in a stratified field.

The Boussinesq approximation is introduced in the governing equations: the fully3D momentum and continuity equations, and the diffusion equations fortemperature and salinity. In the present study, however, the effect of stratificationis negligible and the momentum and continuity equations only are solved over theflow field. The curvilinear grid system is used for fitting not only to the topographybut also to the water-surface. The governing equations are discretized using thefinite-differencing scheme that employs an explicit time-marching procedure with theSGS turbulence model for stratification (Schmidt and Schumann, 1989).

The movement of the water-surface is updated using the ALE method (Hirt et al.,1972). The pressure and velocity are solved by means of a simultaneous iterativemethod, the so-called HSMAC-type algorithm by Hirt et al. (1975). The convectiveterms are discretized using the CIP method (Yabe et al., 1990).

The details of the solution algorithm are explained in Nakayama andSatoh (1999).

4. Computational results and discussions

4.1. TJG model

In the design of a TJG, flow separation due to sudden contraction and expansionnear the entrance is the primary physical feature. For analytical simplicity, the barriersare assumed to be thin and vertical. As depicted in Fig. 1, the TJG model is installed atthe head of a rectangular bay. The water depth h is assumed to be shallow andconstant. The coordinate system has its origin at the center of the inlet. The head of bayis a reflecting wall and no-slip boundary condition is imposed there. Tidal forcing isgiven by imposing a sinusoidally varying water level from Eq. (13) at the entrance ofbay. It is assumed that the incoming tide is sufficiently long compared to the depth andthe width of bay, has small amplitude, and is normally incident to the entrance of bay.Thus, the long tidal waves can be regarded as shallow water waves of celerityCW ¼

ffiffiffiffiffigh

p. Furthermore, wave reflections are assumed insignificant in the system.

During both phases of the tide, strong and uni-directional jets are formed near theinlet, due to water level difference between the inner and outer sides of breakwater.At one time, the inlet acts like a source to one side and a sink of equal magnitude tothe other side. This locally induced jet flow can decrease the residence time of a

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Fig. 1. A tidal-jet generator (TJG) model.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181904

pollutant distributed around the outside of inlet on the ebb tide. In other words, theTJG utilizes tidal energy for water purification in local marine environment byproviding a flushing mechanism.

In such a system, two parameters can be used to characterize both experimentaland numerical results. One parameter is the opening ratio e0 defined as e0 ¼ eh=S,where e is the opening between the adjacent slots of barrier and S( ¼ B�L) thehorizontal area inside the breakwater. Here, the horizontal area and the water depthare important parameters in deciding the optimal opening ratio of the system. Theother important parameter is the maximum intensity of inlet velocity Umax, which isnewly defined here as Umax ¼ U=sA with U ¼ maximum inlet velocity.

4.2. Numerical convergence tests

First, we carried out the numerical convergence tests of the NS–MAC-TIDEsimulation technique in a rectangular bay with its length and width l ¼ 8:75m and

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b ¼ 10m, respectively. Computations were made over a range of CFL numbersdenoted by CW Dt=Dz. All the other parameters were fixed constant, i.e. T ¼ 390 s,2A ¼ 0:0037m, S ¼ 2:01ð¼ 1:15� 1:75Þm2, h ¼ 0:0981m, and e ¼ 0:05m. Sincethe numerical results were not affected significantly by horizontal spacing, thesame grid spacing, Dx ¼ Dy ¼ 0:05m, was used in the horizontal plane for all testcases.

Fig. 2 shows the results of the convergence tests, where NZ indicates the number ofcell used in the vertical direction between water surface and seabed. The maximuminlet velocity was obtained from the time series of the jet velocity probed at theposition of ðx; yÞ ¼ ð0; 0:025mÞ on the water surface during the third tidal motion. Inthe figure, the maximum inlet velocity converges as CFL number decreasesregardless of the number of cells NZ in the vertical direction ranging between 5and 15. It is evident that the CFL number based on the vertical grid spacing shouldbe chosen below 0.5. Since 5 cells in the vertical are shown to be sufficient with CFLnumber smaller than 0.5, we perform all the ensuing calculations under thiscondition.

Figs. 3 and 4 show that the maximum inlet velocity increases linearly with tidalamplitude and horizontal area of TJG.

CFL number based on vertical spacing

Max

imu

m in

let

velo

city

(m

/s)

0 1 2 3 4 50.01

0.02

0.03

0.04

NZ=15NZ=10NZ=5

Fig. 2. Convergence tests for the numerical model, NS–MAC-TIDE model.

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Amplitude of tide (m)

Max

imu

m in

let

velo

city

(m

/s)

0 0.001 0.002 0.003 0.004 0.0050

0.01

0.02

0.03

0.04

Fig. 3. Computed maximum inlet velocity versus amplitude of tide by the NS–MAC-TIDE model.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181906

4.3. Comparison with other numerical methods

Experimental results from previous study (Furukawa et al., 1994) were re-analyzedand a new relationship between the non-dimensional maximum intensity of inletvelocity Umax and the opening ratio e0 was developed. The re-analyzed experimentaldata is presented in Fig. 5. The empirical function Umax ¼ a=e0 with a ¼ 4=3 wasfound to fit the experimental data well. The coefficient a represents the ratio of Umax

to the case where no tidal jet is present, i.e. the value of a shows that the maximumintensity of inlet velocity induced by the tidal jet becomes 4/3 times larger than thatwithout tidal jet. The results computed by the present NS–MAC-TIDE are plotted inFig. 5 with the results of the other three independently developed numerical modelsfor comparison. The specific conditions for the three numerical models are listed inTable 2. In Fig. 5, it is apparent that the two fully 3D numerical models, NS–MAC-TIDE and CIP-CSF, can better predict the maximum intensity of inlet velocity thanthe simpler 3D-ADI and DVM models when compared with the best-fit empiricalfunction. The 3D-ADI and DVM models produce reasonable results for largeopening ratio but deviate appreciably from the experimental data for small openingratio (e0o0:005). The deviation of the 3D-ADI results is more pronounced. Whenthe opening ratio becomes smaller than 0.005, the water level differences between theinner and outer sides of reservoir increase and the 3D effects, such as large water

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Horizontal area of TJG (m2)

Max

imu

m in

let

velo

city

(m

/s)

1 2 3 4 50

0.01

0.02

0.03

0.04

Fig. 4. Computed maximum inlet velocity versus horizontal area of the TJG by NS–MAC-TIDE model.

++ +

+

+

Opening ratio

Max

imu

m in

ten

sity

of

inle

t ve

loci

ty

0 0.005 0.01 0.015 0.020

200

400

600

800

1000

1200

Experimentswithy=a/x,a=4/3DVMCIP-LES-SFNS-MAC-TIDE3D-ADI+

Fig. 5. Relationship between maximum intensity of inlet velocity and opening ratio of the TJG.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–1918 1907

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Table 2

Computational conditions for numerical models

3D-ADI DVM CIP-CSF

Dx ¼ Dy ¼ 0:05m;Dz ¼ h=5m 578 vortex points positioned at

the boundary and 1020 vortex

points released into flow

Dx ¼ Dy ¼ 0:025m (for minimum

size), Dz ¼ h=5m

CFL number ¼ 0.65 Dt ¼ 1 s CFL number ¼ 0.43

Horizontal and vertical eddy

diffusivity, nh ¼ 3.0� 10�2,

nz ¼ 3.0� 10�4, respectively

Horizontal eddy viscosity

coefficient, KX ¼ 0.05 hU

Kinematic viscosity,

n ¼ 1:0� 10�6

Bottom friction coefficient,

gb ¼ 0:05Bottom friction coefficient,

gb ¼ 0:4Bottom friction; No-slip

condition

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181908

falling down and splashing up by tidal jets in the vicinity of the opening, cannot beignored. Then the hydrostatic assumption for pressure in the vertical would nolonger be allowed.

4.4. Flow characteristics of the tidal jets induced by a TJG

The tidal jets are of significant importance in near-shore zone because theyinfluence the mass/sediment transport and the spreading of nutrients. However,there have been very few investigations using complex 3D simulation tools; instead,simplified models have been used under the assumption of steady jet and hydrostaticpressure in the vertical direction. In this section, the 3D flow characteristics of thetidal jets and their effects will be discussed based on accurate simulations by theNS–MAC-TIDE.

Fig. 6 shows the depth-averaged momentum distribution of residual flow for threedifferent openings e ¼ 5, 15 and 25 cm, corresponding to the non-dimensional valuese0 ¼ 0:002453, 0.007358 and 0.012263. The depth-averaged momentum was obtainedby taking the absolute value of the residual flow velocity remained during the thirdtidal motion. As the opening gets smaller, the momentum is distributed farther andmore widely in the vicinity of the TJG, in which high values represent the area ofviolent tidal jet motions.

In order to observe a time-sequential process of the tidal jet in more detail, a seriesof snap shots of the case e ¼ 5 cm are presented in Fig. 7. The pictures exhibit thecontour maps of tide-induced jet velocity and jet-induced vorticity during a tidalmotion, in which the dotted lines indicate the jet velocity and the coloring representsthe vorticity near the water surface. Fig. 8 shows the corresponding figures for themomentum. Each snap shot is made for one-eighth of the tide period. During bothphases of tide, the typical processes of the tidal jet are well reproduced numerically.Particularly, the jet issued from the inlet induces a pair of large-eddy vortex,entraining the surrounding waters as it proceeds far from the inlet. The jets interactwith the vortices causing the dissipation of their momentum gradually.

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X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

t=(1/8)T t=(2/8)T t=(3/8)T t=(4/8)T

t=(8/8)Tt=(7/8)Tt=(6/8)Tt=(5/8)T

ε=5(cm)

Fig. 7. Time-sequential development of tidal jet velocity (dotted lines, interval 0.004m/s) and jet-induced

vortex (flood, blue: 0.12/s, red: �0.12/s).

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4ε=5(cm) ε=15(cm) ε=25(cm)

Fig. 6. Momentum distribution of residual flow with the three different openings; blue: 0 and red: 0.005m/s.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–1918 1909

Fig. 9 shows the corresponding time sequences of water-surface configuration. Thelocal waves are generated through the inlet, and then propagated and dissipatedindependently with the tidal motion. It may affect the vertical structure of the flowfield, even though its amplitude and length are much smaller than those of tide.

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X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

X (m)

Y (

m)

-1 -0.5 0 0.5 1

-1

0

1

2

3

4

ε=5(cm) t=(1/8)T t=(2/8)T t=(3/8)T t=(4/8)T

t=(8/8)Tt=(7/8)Tt=(6/8)Tt=(5/8)T

Fig. 8. Time-sequential development of momentum around a TJG (blue: 0.0m/s, red: 0.012m/s).

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181910

In Fig. 10, the vertical distribution of centerline velocity is exhibited as timemarches, in which the reflecting wall is located at the left side. For this case, the localReynolds number at the inlet caused by the induced jet flow is around 7.5� 102. Ifthe jet flow can be treated as laminar flows, the boundary layer thickness from theBlasius solution is approximately 5.8 cm at the position of y ¼ 2:0m, and theboundary layer occupies about 60% of the water depth.

The vertical structure of velocity and the development of boundary layer nearseabed are closely related to the sediment transport. They are especially important topredicting the distribution of shear stresses for the transport mechanism. Fig. 11shows the distribution of shear stress along the centerline on seabed for e ¼ 5 and15 cm. Here, the shear stress was calculated by

tBr¼ ðnþ nSÞ

qV

qz. (20)

In the figure, the maximum strength of shear stress occurs when the jets areformed from the inlet at both t ¼ ð1=4ÞT and (3/4)T. At the moment, the maximumshear velocity converted from bed shear stress is estimated at 0.35 cm/s. It is causedby the jet formation around the inlet. The corresponding limitations of grain size forsuspended load and bed load are approximately 0.065 and 0.0055mm, respectively(Nichols and Biggs, 1985). From this calculation, the sand drift in suspension may beeasier to occur than that by bed load under the present situation.

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Fig. 9. Time-sequential movement of water surface, 40,000 times enlarged on vertical scale.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–1918 1911

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Y (m)-1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 2

-0.0981

0.0000

Z (

m)

Z (

m)

Z (

m)

Z (

m)

Z (

m)

Z (

m)

Z (

m)

Z (

m)

-0.0981

0.0000t=(1/8)T

t=(2/8)T

t=(3/8)T

t=(4/8)T

t=(5/8)T

t=(6/8)T

t=(7/8)T

t=(8/8)T

-0.0981

0.0000

-0.0981

0.0000

-0.0981

0.0000

-0.0981

0.0000

-0.0981

0.0000

-0.0981

0.0000

Fig. 10. Time-sequential development of vertically distributed centerline velocity computed by the

NS–MAC-TIDE model.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181912

ARTICLE IN PRESS

ε=5(cm)ε=15(cm)

t=(4/4)T

ε=5(cm)ε=15(cm)

t=(3/4)T

ε=5(cm)ε=15(cm)

t=(2/4)T

X (m)

Sh

ear

stre

ss

-1 0 1 2

X (m)-1 0 1 2

X (m)-1 0 1 2

X (m)-1 0 1 2

-1.5E-05

-1E-05

-5E-06

0

5E-06

1E-05

1.5E-05

Sh

ear

stre

ss

-1.5E-05

-1E-05

-5E-06

0

5E-06

1E-05

1.5E-05

Sh

ear

stre

ss

-1.5E-05

-1E-05

-5E-06

0

5E-06

1E-05

1.5E-05

Sh

ear

stre

ss

-1.5E-05

-1E-05

-5E-06

0

5E-06

1E-05

1.5E-05

ε=5(cm)ε=15(cm)

t=(1/4)T

Fig. 11. Time-sequential distribution of shear stress along centerline seabed.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–1918 1913

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J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181914

4.5. Numerical assessment of the flushing system for water purification

Finally, to investigate the flushing mechanism for water purification and masstransport in front of the TJG, the time-residual rates of concentration are calculatedfor three openings of e ¼ 5, 15 and 25 cm. For this, the following scalar transportequation of concentration C was used:

qC

qtþ rðu � CÞ ¼

nþ nSSCH

� �r 2C. (21)

In Eq. (21), SCH is the turbulent Schmidt number chosen as 0.7 (Rodi, 1993). Theconcentration C can be pollutants, water nutrients, or littoral effluent materials, etc.The concentration field was introduced after the second tide, and then computedwith the flow field simultaneously at each time step in the time-marching procedure.Initially the concentration is set at C ¼ 1 in the rectangular region located in front ofthe TJG and at C ¼ 0 otherwise (see Fig. 12). The time rate of concentration residualis calculated by integrating the concentration residual over the whole monitoredcontrol volume chosen as 2hm3 equal to the enclosed volume of the TJG, 2m3. Thenon-dimensional rates are normalized by initial values. The time variation ofconcentration residual is shown in Fig. 13. During four tidal periods afterintroducing the concentration, the total proportions of concentration in the control

X (m)

Y (

m)

-2 -1 0 1 2

-1

0

1

2

3

4

5

C=1.0

C=0.0

C=0.0

Controle Volume

Fig. 12. Initial condition for the calculation of concentration transport.

ARTICLE IN PRESS

time (sec)

Res

idu

al o

f co

nce

ntr

atio

n

1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε=5(cm)

ε=25(cm)

ε=15(cm)

47.6%

75.2%

84.7%

Fig. 13. Time evolution of the residual of concentration in a monitored control volume.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–1918 1915

volume are decreased by 52%, 25%, and 15% for e ¼ 5, 15, and 25 cm, respectively.From these results, we can approximately predict the residual time for threeopenings as 8T, 16T and 27T, respectively. Fig. 14 shows the instantaneouslydistributed concentrations on the water surface at four different time steps and forthree opening ratios. In the case of e ¼ 5 cm, the concentration is remarkablydecreased and flushed away from the inlet as time increases, while the effects are lessobvious in the other cases. Through these examples, it is clearly verified that thelocally induced jet flows can decrease the residual time of a pollutant distributedoutside the inlet, which illustrates the beneficial utilization of tidal energy for waterpurification in local marine environment by artificially providing a flushingmechanism. More large-scale experiments are needed in the future to further verifythe present numerical study.

5. Concluding remarks

The flow characteristics of tidal jets induced by the Tidal-Jet Generator (TJG) areinvestigated using a finite-difference numerical scheme, named Navier–Stokes(NS)–MAC-TIDE, based on the fully 3D NS equations.

The fully 3D NS–MAC-TIDE simulations are compared with the experimentalresults of Furukawa and three other independently developed numerical models, 3D-ADI, DVM, and CIP-CSF. These models are based on quasi-3D, 2D depth-averaged, and fully 3D NS equations, respectively. It is apparent that the present

ARTICLE IN PRESS

X (m)

Y (

m)

-2 -1 0 1 2X (m)

-2 -1 0 1 2X (m)

-2 -1 0 1 2X (m)

-2 -1 0 1 2

X (m)-2 -1 0 1 2

X (m)-2 -1 0 1 2

X (m)-2 -1 0 1 2

X (m)-2 -1 0 1 2

X (m)-2 -1 0 1 2

X (m)-2 -1 0 1 2

X (m)-2 -1 0 1 2

X (m)-2 -1 0 1 2

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5Y

(m

)

-1

0

1

2

3

4

5Y

(m

)

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5

Y (

m)

-1

0

1

2

3

4

5ε=5(cm) t=3.000T ε=5(cm) ε=5(cm) ε=5(cm)

ε=15(cm)ε=15(cm)ε=15(cm)ε=15(cm)

ε=25(cm) ε=25(cm) ε=25(cm) ε=25(cm)

t=4.000T t=5.000T t=6.000T

t=3.000T t=4.000T t=5.000T t=6.000T

t=3.000T t=4.000T t=5.000T t=6.000T

Fig. 14. Distribution of the residual of concentration at various time for the cases of e ¼ 5, 15 and 25 cm.

J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–19181916

fully 3D numerical model, NS–MAC-TIDE, can better predict the maximumintensity of inlet velocity than the other numerical models when compared with thebest-fit empirical function from Furukawa’s experiment. It is also seen that thenarrower opening is more effective in producing higher momentum and strongertidal jets.

The numerical simulations using NS–MAC-TIDE can successfully reproduce theprocesses of generation, development, and dissipation of tidal jets. Using the 3Dnumerical models, the actual velocity profiles and shear stresses of the tidal jet flowsare calculated so that the corresponding mass or sediment transport can bereasonably assessed. Through numerical assessment, it is verified that the residualtime of a pollutant distributed around the front of the TJG can be decreased

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J.-C. Park et al. / Ocean Engineering 33 (2006) 1896–1918 1917

significantly by the locally induced tidal jet, i.e. the proposed TJG can beneficiallyutilize tidal energy for water purification in local marine environment by providing aflushing mechanism.

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