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WATERWindows Based Drainage Design Software

M C Zhou (cmczhou@ntu.edu.sg)Tommy S W Wong (cswwong@ntu.edu.sg)

Introduction

As Singapore is a tropical country with abundant rainfall,in order to minimize the risks of flooding and the associatedhealth hazards, it is important that an effective drainagesystem be put in place. To this end, the drainage systemsin Singapore are required to be designed in accordancewith the “Code of Practice on Surface Water Drainage”(Public Utilities Board, 2000). In this Code, the drainagesystems are required to satisfy the Rational formula andthe Manning equation. As such, designers are often requiredto carry out a tedious and time consuming iterationprocedure. After satisfying the Rational formula and theManning equation, the design may still require revision asit is also obliged to satisfy other requirements as specifiedin the Code. With the intention of saving designers muchtime and effort, a Windows-based drainage design softwarehas been developed. The software is known as the “DrainageDesign Software” or “DDSoft”. This article describes thefeatures in the DDSoft.

Features of DDSoft

The DDSoft is written in Visual Basic language, andoperates on the Microsoft Windows platform. It comprisesthree main interfaces: one principal and two supplementary.

In the DDSoft, the input data and the output results aredisplayed on the same principal interface, namely “DrainageDesign Software” as shown in Figure 1. This interfaceenables the designers to carry out a design easily. At thetop left hand corner, the designer can enter a project title.Then he is required to select a design option. Two optionsare available: (i) channel, (ii) sewer. For each design option,there are three input modes. For the “channel” option, theinput modes are: (i) channel bed slope and channel width,(ii) channel bed slope and flow velocity, and (iii) channelwidth and flow velocity. For the “sewer” option, the inputmodes are: (i) sewer bed slope, (ii) sewer diameter, and(iii) sewer flow velocity. The designer then selects therainfall location, either “Singapore” or “others”. If heselects “Singapore”, he can then select a return period froma drop down menu. Alternatively, he can enter the returnperiod directly. On the other hand, if he selects “others”,the designer is required to input the rainfall intensity-duration equation of that location.

After entering the information on the rainfall, the designeris required to enter the information on the catchment. Froma drop down menu, the designer may select a singledevelopement type. For this case, the value of the runoff

coefficient according to the Code will appear automatically.Alternatively, the designer may select the compositedevelopment type. For this case, as shown in Figure 2, asupplementary interface known as the “CompositeDevelopment Catchment” will appear. The designer mayselect the appropriate development type in terms of area inha, or in %. When the designer clicks the button “Apply”,DDSoft will work out the weighted runoff coefficient. Ifthe designer is satisfied with his input, he can click “OK”and DDSoft will return to the principal interface. If thepercentage for any development type or the summation ofthe percentages exceeds 100%, DDSoft will give a warningmessage. The designer is then required to amend his input.In the “Composite Development Catchment” interface, ifthe designer enters the area in terms of ha, the total areawill be calculated and automatically transferred to the

Figure 1. Principal interface for data input and output results

Figure 2. Supplementary interface for calculatingweighted runoff coefficient

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principal interface as the catchment area. Otherwise, on theprincipal interface, the designer is required to enter thecatchment area. The designer can specify his own runoffcoefficient value if it is different from those described inthe Code. Other information on the catchment includes theoverland flow time and the upstream channel (or sewer)flow time. Their values can be specified directly by thedesigner, or computed using the equations provided.

Next, the designer is required to enter the information onthe channel or sewer. For the “channel” option, the designeris required to select a channel shape. Four channel shapesare available. They are vertical curb, triangular, rectangularand trapezoidal. For the “sewer” option, one shape isavailable and it is circular under full flow condition. Afterselecting the channel shape, the designer is required toselect a surface type. From a drop down menu, the designermay select a single surface type. For this case, the value ofthe Manning’s roughness coefficient according to the Codewill appear automatically. Alternatively, the designer mayselect a composite surface type. For this case, as shown inFigure 3, a supplementary interface known as the“Composite Surface Channel/Sewer” will appear. Thedesigner may select the appropriate surface type in termsof wetted perimeter in m, or in %. When the designerclicks the button “Apply”, DDSoft will work out theequivalent Manning’s roughness coefficient. If the designeris satisfied with his input, he can click “OK” and DDSoftwill return to the principal interface. If the percentage forany surface type or the summation of the percentages exceed100%, DDSoft will give a warning message. The designeris then required to amend his input. If the channel (orsewer) surface type is different from those described in theCode, the designer can specify his own value directly. Theremaining input data on the channel (or sewer) depends onthe selected design option and input mode, and the channel

Figure 3. Supplementary interface for calculating equivalentManning’s roughness coefficient

shape. The requested data will be a combination of channellength, channel width (or sewer diameter), channel bedslope (or sewer bed slope), flow velocity, and channel sideslope. DDSoft caters for channels with unequal side slopes.

With the input data, DDSoft will perform the designcalculations as soon as the designer clicks the button “Run”.The output results are displayed on the principal interface.The output results include time of concentration, designrainfall intensity, design discharge. The remaining outputresults depend on the selected design option and input mode.They can be the channel flow time, flow velocity (or sewerflow velocity), Froude number, channel width (or sewerdiameter), or channel bed slope (or sewer bed slope). Forthe output results, DDSoft will perform an automatic checkto see if they satisfy the other requirements in the Code(e.g. minimum and maximum design velocities, andmaximum Froude number). If there is any result that doesnot satisfy the requirements, DDSoft will display a warningmessage, and give a suggestion on how to revise the design.For certain sets of input data, there are no possible solutionsfor flow depth or channel width. For these cases, DDSoftwill also display a warning message and a suggestion onhow to revise the design. The input data and the outputresults can be printed out as reports or saved as ASCIIfiles. The latter output option allows earlier designs to beretrieved easily.

Conclusion

The “Drainage Design Software” or “DDSoft” is aWindows-based software. It is able to perform the drainagedesign calculations according to the “Code of Practice onSurface Water Drainage”. In the DDSoft, the input dataand the output results are displayed on the same principalinterface. This enables the designers to carry out a designeasily. Further, for certain sets of input data, if the outputresults do not satisfy the requirements in the Code, DDSoftwill display a warning message with suggestions on howto revise the design. The input data and the output resultscan also be printed out as reports or saved as ASCII files.The latter output option allows earlier designs to be retrievedeasily. DDSoft will save the designers a lot of time andeffort in carrying out a drainage design. DDSoft is availableon the web site: http://www.ntu.edu.sg/home/cswwong/software.htm.

Reference

[1] Public Utilities Board (2000). “Code of Practice onSurface Water Drainage.” Drainage Department,Singapore.

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Influence of Tides on aCoastal Aquifer in Singapore

Mzila N (PZE348703@ntu.edu.sg)Shuy E B (cshuyeb@ntu.edu.sg)

Introduction

The periodic rise and fall of tides in the open sea producesinusoidal fluctuations in the groundwater level in an adjacenthydraulically connected coastal aquifer. In an unconfinedaquifer, the fluctuations occur as pressure waves generatedby changes in storage due to dewatering and re-saturation ofpores propagate inland. The characteristics of these pressurewaves, i.e. their velocity, amplitude, wavelength andattenuation, are a function of the tidal period and amplitude,the aquifer’s transmissivity and storage coefficient, and thedistance inland.

The present study was conducted in the shaded areas (A4and A2) of the Changi reclaimed land as shown in Figure 1in local co-ordinates. The fluctuations in groundwater levelat the site caused by tidal influence were measured using aTROLL 8000 pressure transducer at observation wells A2S10,A4S17, A4S08, A4S20, A2S09 and A2S08, shown in Figure2. An isometric section of the study area is shown in Figure3.The aquifer is composed of artificial sand fill, with meangrain size 0.75 mm, porosity n = 40, d10 = 0.12 to 0.37 mm,shell content = 6 to 9%, uniformity coefficient = 2.5 to 4,and extends from the ground surface to a depth of between12 and 16m.

Relation between tide level and piezometric level

The observed groundwater level fluctuations can be dividedinto two components: (1) tidally induced fluctuations, and(2) fluctuations caused by other factors, and can be describedby the following equation (British Geological Survey, 2001):

(1)

where:h’ = groundwater elevation without tidal influence (L)h = observed groundwater elevation (L)H = tidal elevation on surface water body (L)Etide = tidal efficiency (dimensionless)T = the time when groundwater elevation was

measured (T)tlag = time lag between tidal effects in surface water

body and corresponding effects at groundwaterobservation points(T).

The relationship between tidally induced piezometric levelsand tidal fluctuations is determined by two parameters: tidalefficiency (Etide), and time lag (tlag). The tidal efficiency factoris the ratio of amplitudes of the piezometer readings andtidal readings. The time lag is the time difference betweenthe tidal levels and the corresponding groundwater levels.

Figure 3. An isometric section of the study area

Figure 1. Study area (shaded) within Changi reclaimed land

Figure 2. Distribution of installed piezometers at the study area

In a confined, homogenous and isotropic aquifer, sinusoidaloscillations in pressure will propagate along the aquiferaccording to the following equation (Erskine, 1991):

(2)

where h = groundwater head relative to mean sea level (m);x = distance from the sea (m); t = time (d); to = period oftidal oscillation (d); ho = amplitude of tidal oscillation (m);T = transmissivity of aquifer (m2/d); and S = storage ofaquifer (dimensionless).

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Both the tidal efficiency factor and time lag are determinedby a number of factors including aquifer hydraulicconductivity and storativity (or diffusivity), aquifer thickness,and distance of the observation piezometer from the sea.The solution of equation 2 shows that the tidal oscillationsremain sinusoidal with a time lag and decrease exponentiallyin amplitude with distance from the sea. The magnitudes ofthese parameters are as follows:

(3)

and

(4)

where K = aquifer hydraulic conductivity (L/T) and b =aquifer thickness (L).

Equations 3 and 4 developed for confined conditions canalso be used for unconfined conditions if the range offluctuation of groundwater levels is small compared to thesaturated depth of the aquifer.

Calculation of tidal efficiency and time lag

For calculating the tidal efficiency factor and time lag fromthe observed tidal and tidally induced groundwater piezometerdata, an observation period is selected during which otherfactors affecting groundwater levels such as rainfall arenegligible, i.e. (dh’/dt=0). The groundwater level fluctuationsare then solely a function of tidal fluctuations, and equation1 reduces to :

(5)

For a time period from t0 to t1 in the groundwater observationrecord, the solution of equation 5 can be obtained byintegration as follows :

(6)

The above integral can be expressed as follows:

(7)

From equation 7, the tidal efficiency for the period from t0

to t1 can be calculated as follows:

(8)

High and low tides are selected from tidal records and thecorresponding groundwater high and low levels are identified.The time lag is calculated from:

(9)

whereti(tide) = Time for the ith high (or low) tide (T)ti(gw) = Elevation time for the ith high (or low)

groundwater elevation corresponding to the ith

high (or low) tide (T).

and the tidal efficiency is calculated from :

(10)

whereHi = ith high (or low) tidal elevation (L)hi = The ith high (or low) groundwater elevation

corresponding to the ith high (or low) tide (T).

The time lag and tidal efficiency factor were calculated for6 groundwater piezometers at the study site. The time lagwas calculated by a least-squares fit method. In order toperform this analysis, the time series of piezometric readingswere compared to tidal elevations for the same period andthe piezometric series were shifted until the sinusoidalfunction was synchronous with tidal elevation data. The shiftrepresents the time lag between observed groundwaterpiezometric data and tidal data. The piezometric readingswere then shifted in elevation to have the same mean as thetidal records and then amplified by the tidal efficiency factorfor each standpipe. Mathematically, this shift andamplification may be written as:

(11)

where= piezometric reading at time t (m);= shifted piezometric level at time t (m);= mean piezometric level (m);= mean tidal level (m);= tidal efficiency factor (dimensionless).

The answers were checked visually by overlaying the shifted,amplified and correctly lagged plot with the tidal record, asshown Figure 4.

Results and discussion

The time lags and log tidal efficiency factors were plottedagainst distance from the coastline for the groundwaterpiezometers used in the study as shown in Figures 5 and 6.From these graphs it can be observed that time lags vary

Figure 4. Piezometer A2S10-tide with shifted, amplified, andlagged (175 minutes) piezometer levels

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Figure 5. Time lag versus distance from the Sea

Figure 6. Log tidal efficiency versus distance from sea

linearly with distance from the coastline and the tidalefficiency factor varies exponentially as predicated inequations 3 and 4 respectively. Regression analyses wereperformed on the data points to obtain the slopes of the best-fit lines as follows:

Slope = 1.8 x 10-3 day/m (time lag) (Figure 5)Slope = -1.23 x 10-2 /m (tidal efficiency) (Figure 6)

Using equations 3 and 4, the K value for the study site isestimated at 136 m/d using the time lag method, and430 m/d using the tidal efficiency method.

References

[1] British Geological Survey, 2001, “Review ofGroundwater and Salinity Management inUnconsolidated Sedimentary Aquifers”, Review NoSWR0001.SIWIN

[2] Erskine, A.D., 1991, “The Effect of Tidal Fluctuationon a Coastal Aquifer in the UK”, Ground Water, 29,pp.556-562

Three Dimensional Flow Prediction usingPOM for Waters off Singapore

Arthur Lim (ctblim@ntu.edu.sg) Edmond Lo (cymlo@ntu.edu.sg)

Tan Soon Keat (ctansk@ntu.edu.sg)

Introduction

The numerical Princeton Ocean Model (POM) was applied to a3D flow prediction for waters off Singapore. This was done aspart of an ongoing 3-year joint MPA-NTU project to develop andvalidate POM into a Computational Ocean Modelling System(COMS) for tidal and circulation flow predictions for the SouthChina Sea. The present work reported here forms the initial partof the effort with subsequent work focusing on developing aSouth China Sea model and a graphical user interface (GUI)platform for COMS.

The POM prediction for waters off Singapore was performed ona small-scale domain stretching from 103.4O E to 104.3O E inlongitude and 1O N to 1.4O N in latitude. Excellent predictionswere obtained, specifically, the variation of the predicted tidalcurrent magnitude and direction at 3 different locations for 2different time periods compared very well with MPA field data

taken at these locations. Vertical profiles of current characteristicswith depth variation on these locations also showed the ability ofthe POM to model well the physical vertical structures ofdownwelling and direction reversal induced by bathymetry change.

Brief description of POM

POM is a three-dimensional, time-dependent primitive equationmodel, first created by Blumberg and Mellor [1] around 1977,and has now been extended and used by over 450 research groupsworldwide for applications ranging from estuarine, to coastal andto oceanic bodies.

The principal features of the model incorporates the following:

• A bottom-following sigma coordinate scaling for the verticaldimension to model significant bottom topographicalvariability.

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• An imbedded second moment turbulence closure model ofMellor and Yamada [2] to provide a realistic parameterizationof the vertical mixing process.

• A curvilinear orthogonal coordinate and the staggered“Arakawa-C” differencing on the horizontal grid. Thehorizontal diffusivity coefficients are based on Smagorinsky[3] parameterisation.

• A free sea-surface coupled with a split-mode time step. Inthis treatment, an external 2D depth averaged mode is used,solving first for the sea-surface elevation and depth-averagedvariables, and uses a smaller time step based on CFLrequirement and the external surface wave speed. Theinternal mode is 3D and solves for the vertical and horizontalvelocities, temperature and salinity through the layers, anduses a longer time step due to implicit time stepping and aCFL condition based on the internal wave speed.

Governing equations

The equation set of POM, consists of continuity, momentumtransport, energy or temperature conservation and salinityconservation. These are written, in a physical coordinate systemwith x increasing eastward, y increasing northward and z increasingvertically upward, and with corresponding velocities (U,V,W), as :

Continuity: (1)

Momentum :

(2a,b,c)

Here ρo is the reference density, ρ the in-situ density, g thegravitational acceleration, P the pressure, KM the vertical eddydiffusivity of momentum mixing, and f the Coriolis parameter,given in terms of Earth’s latitude by the β-plane approximation,2Ω sin(φ). Fx and Fy are terms describing the horizontal mixingprocesses, and in analogy to molecular diffusion, are written as :

(3)

AM is the horizontal diffusivity computed via Smagorinsky’sparameterization based on the local grid scales and deformationfield. In present computations, AM is generally initialized to bebetween 2 to 20 m2/s.

Given that the free surface is expressed by η(x,y,t), as depictedbelow, the pressure at a given depth z can be obtained byintegrating Eqn(2c) over z to give

(4)

where atmospheric pressure Patm is assumed to be constant.

The temperature (θ) and salinity (S) conservation equations arewritten in similar forms as the U,V momentum equations. Forour present case of Singapore waters, whereby both fields are

generally uniform due to the small water depth (averagingapproximately 35m), these are not solved, even though they arepart of the POM model. In deeper waters with large geographicalcoverage such as the South China Sea, these terms would influencethe pressure and density fields, through the baroclinicapproximation where the density is given as ρ = ρ(θ,S,p). Thiswould in turn affect the velocity fields in Eqn (1) and (2) throughthe density term and pressure gradient terms.

In Equation (2), the vertical mixing coefficient KM is based onthe second-order turbulence closure model of Mellor and Yamada[2]. It characterizes the turbulence by 2 equations for turbulencekinetic energy, q2/2, and a turbulence macroscale, l. Furtherdetails can be found in references [1] and [2]. In particular KM

is expressed as a function of l and q.

The formulation in physical x,y,z coordinates has disadvantagesof numerical implementation and accuracy in vicinities of largebathymetric variations. In POM, this is overcome through theuse of the sigma coordinate that transforms the sea surface andthe bottom into coordinate surfaces as

(5)

where H(x,y) is the bottom bathymetry and η(x,y,t) is the surfaceelevation. Thus the sea-surface at z=η is mapped to σ =0 and thesea-bed at z=−H is mapped to σ=-1. After conversion, the velocitycomponent normal to sigma surfaces ω is related the Cartesianvertical velocity, W as:

(6)

The POM equations have further been modified to include theequilibrium astronomical tide level at any geographical positionand time. These relationships are based on Newtonian theory oftide generation from the gravitational and centrifugal forces ofEarth-Moon and Earth-Sun System. This astronomical tidecomponent is built into our code as an additional source of forcingfunction to account for large astronomical gravitational forcesover large geographic surface.

Vertical boundary conditions

The boundary conditions at surface and bed are:

(7a,b)

The surface boundary conditions for (3) and (4) are

(8a,b)

Figure 1. Bottom-following Sigma Coordinate ScalingRepresentation

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where the right hand side of Eqn (8a,b) are the input values of thesurface turbulence momentum flux or the surface wind stress vector.In our application, the simulation was done without surface windforcing. The corresponding bottom boundary conditions are

(9a,b)

(9c)

κ = 0.4 is the von Karman constant and zo is the roughnessparameter, and is taken as 0.01m for the present simulation.

Lateral boundary conditions

On the solid lateral boundaries, i.e. land borders, free-slip conditionbut zero normal velocity are applied. Along open boundarieswhere the numerical grid cells end but fluid motion is unrestricted,tidal elevations are prescribed as either harmonic series or timeseries interpolations as forcing for the surface elevation, η(t).Correspondingly for the velocity fluxes, radiation boundaryconditions are used as,

(10)

where ‘n’ refers to direction to the normal to the boundary, andci is the internal wave speed.

In the present work, over each open boundary, a single set ofmeasured time-series tidal elevation data, as measured at locationalong the boundary, is set as the sole forcing function on thatboundary. The numerical boundary treatment for the sea-levelinputs has further been modified to accommodate (a) time-seriesamplitude-time input and (b) harmonic amplitude-phase inputfrom field tidal station. For the Singapore water case, theharmonic component input focused on only 13 significant modes,namely the semi-diurnal modes (M2, S2,N2,K2 Tide), thediurnal modes (K1,O1,P1,Q1 Tide) and several Long-PeriodModes and Mixed Modes (MF,MM, SS or SSA, M4 and MS4Tide). These 13 modes typically constituted higher than 94%of the total spectra energy of tides.

Singapore water model and results

Using a combination of two bathymetry maps with resolutionsof 500m and 1000m spacing respectively, a model on theSingapore water was created with the open boundary locationsand geographical domain as shown in Figure 2. This modelconsists of 115*85 IJ horizontal grids and 11 K vertical grids,with an average horizontal grid spacing about 850m and avertical spacing of about 0.1 ∆z/H scale. In the figure, Stations4,5,6 refer to the locations of 3 sites where the MPA tidal currentdata was collected for 2 periods in August and December of1978 at a depth of 10m. This data are used to calibrate andverify the results of our Singapore Water model. Figures 3shows the typical comparisons of the 3D simulation results onthe tidal horizontal current velocities at one of these locationsfor the December period. It shows a very good correlation atsigma layer 4 (depth near 10m) with the measured data.

It is noted that the depth-averaged values, computed using theexternal mode portion of the POM model, give a poorer

mode is essentially a 2D computation that integrates over thevertical layers along the water column to give a single depth-averaged horizontal flow vector value. The external mode portionin the POM model is, however, essential in that it is used tosolve for the sea-surface elevations into which the 3D internalmode computation is coupled.

Besides being able to predict the horizontal structures, the POMmodel is also able to capture the vertical structure. Figure 4shows the vertical profiles of the computed velocity componentsat the 3 locations for a specific time. It is interesting to observethat at Station 5 near the greatest depth location in the SingaporeStrait as seen in Figure 2, the vertical velocity profile revealsa noticeable downwelling as compared with the other 2 locationswith moderate depths with values nearer to the average depthrange of 35-45m. The 3D computations also reveal furthervariation of the current over the internal layers, such as a

Figure 2. Domain Area and Open Boundary Location of theSingapore Water Model, showing the bathymetry contour.

Figure 3. Comparison of Tidal Current’s Magnitude andDirection between computation and data at Station 6

(at 104º9’42” E and 1º17‘27”N)

prediction of the tidal current as compared with the 3D internalmode calculation over each layer. This is because the external

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direction reversal phenomena, as depicted at Station 6 in Figure4. At this location near to the shoreline, this phenomenon maybe attributed to mechanism associated to tide turning.

Conclusion

The POM has been successfully applied to predictions of thecurrent for waters off Singapore. This initial phase has alsorevealed vertical flow structures in the deeper parts of theSingapore Straits. Work is now focusing on developing asimulation for the South China Sea. As part of this simulation,databases of bottom topography, sea surface elevations, surfacefluxes and boundary forcing are being researched and assembled.

References

[1] Blumberg, A.F., and G.L. Mellor, A description of a three-dimensional coastal ocean circulation model, in Three-Dimensional Coastal Ocean Models, V.4, pp. 208, 1987.

[2] Mellor, G.L, and T. Yamada, “Development of a turbulenceclosure model for geophysical fluid problem”, Rev. Geophys.Space Phys., 20, 851-875, 1982.

Figure 4. Vertical Profiles of Velocity Components for the 3locations at a specific time of 0800 hr on 7th August 1978.

Flow Induced Hydrodynamic Forceson a Slender Ship

Zhou Zexuan (czxzhou@ntu.edu.sg)Edmond Lo Yat Man (cymlo@ntu.edu.sg)

Tan Soon Keat (ctansk@ntu.edu.sg)

Introduction

Computation of ship hydrodynamics has received significantattention for applications to ship maneuverability problems.Maruo and Song (1990) developed a boundary elementsolution for a slender ship based on 3-D Green’s functionsin deep water and using Kelvin sources that satisfy the linearfree surface conditions exactly. The corresponding non-linearwave problem was addressed by Kim and Lucas (1990) usingRankine sources and with iteration on the source distributionand wave profile. More recent work has focused ondeveloping Green’s function in finite water depth (Chen andNguyen, 2000) and using fully computational fluid dynamics(CFD) approaches (Shin and Takanori, 2000). It is well knownthat the hydrodynamic loading changes significantly whenthere is significant blockage of the flow by the hull. Thusthe focus here is on developing a slender 2-D hydrodynamiccode suitable for engineering predictions of the forces on aslender ship moving at a constant speed and an angle ofattack in water of finite depth. Effects of applying the linearversus the full nonlinear surface conditions and finite depthare assessed.

Mathematical model

To develop a mathematical model, we considered a shipmoving with a constant forward speed at an attack angle infinite water depth. In the mathematical model, the ship isfixed in a uniform cross flow with the constant velocityin the horizontal plane. A co-ordinate system is set up withthe origin on the undisturbed free surface, the x- and y- axeson the horizontal plane and the z-axis vertically upwards.The steady water motion around a slender ship is consideredinviscid, incompressible and irrotational, allowing the use ofa total velocity potential as

(1)

where φ and are the induced velocity potential caused bythe ship and position vector respectively. The velocitypotential is governed by Laplace’s equation,

(2)

No flux boundary conditions apply on the rigid ship hull and

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sea bottom. The ship-induced velocity potential satisfies thefar-field conditions of zero-perturbation. As this zerocondition is difficult to simulate, we adopted a less stringentcondition of zero perturbation velocity and the far-fieldboundaries were moved outward until the results converged.The usual non-linear kinematical and dynamic conditionsapply on the free water surface. The linear form based onBernoulli’s equation for pressure is

(3)

The corresponding non-linear form at z = η is given as

(4)

The induced flow around the ship for small attack angles ismade two-dimensional by invoking the slender bodyassumption. Thus the flow in each cross-section depends onthe body shape and the upstream flow through the surfaceconditions, Equations (3) and (4).

Computational method

Slender body assumption is invoked so that Laplace’sequation need only be solved at each cross sectional plane(Figure 1). The boundary element method with constantelements is adopted to solve the flow-field. The normaldirection of the element is assumed to be the normal directionat its mid-point so that numerical difficulties of sharp pointsare avoided. The discretization of flow boundaries is suchthat boundary elements on the hull surface are larger thanthose on the free water surface near the ship. This ensuresthat the simulation is more quickly convergent in the casesconsidered. The derivatives in x in Equations (3) and (4) areapproximated using backward differencing and the centraldifferencing is applied to the other derivatives. Iteration isused for the handling the non-linear terms with anapproximation of using the induced velocity potential andwater surface elevation from the previous cross-section asthe starting estimates. Lastly, Gauss’s quadratic interpolation

is applied to the integrals over boundary elements distributedon the ship hull.

For calculations using the full nonlinear surface conditionEquation (4), iteration is used on both the velocity potentialand water surface elevation. The latter is given by the freesurface dynamic condition. The convergence criterion is thatthe sum of absolute values of difference in the velocitypotential between successive iterations is less than thespecified tolerance of 10-8.

As the ship is fixed in the horizontal plane, only threehydrodynamic forces arise, wave resistance, lateral force andyaw moment. These are evaluated based on pressure integralsover the hull surface. The pressure on the hull body surfaceis determined by the Bernoulli equation

(5)

where ρ is the water density.

Numerical results

The Wigley hull is adopted for presentation of results asprevious experimental results and fully 3-D results areavailable for comparison. Its hull surface is given by

(6)

where l =1m is the half ship length, b = 0.1m is the half shipbreadth and d = 0.125 m is the ship draft. The first casesconsidered are those in deep water. Figures 2-4 show thecomparisons between the present numerical wave profilesand the experimental results from Namimatsu et al (1985),for the cases of (i) F

r = 0.32, α = 0o and (ii) F

r= 0.267, α

=10o. The hydrodynamic forces and yaw moment for differentwater depths, attack angles and waves are also listed in Tables1 and 2. Table 1 shows two cases on using linear wave atzero and 10o attack angles. The results are given ascoefficients where the wave resistance and lateral force arenormalised by and yaw moment by .The results indicate good agreement with past experimentalwater surface profile and force/moment coefficients fromfully 3-D treatments. The results also showed that the non-linearity has a small effect on the water surface displacementbut reduces the wave loading. The attack angle leads to theasymmetry of wave pattern and lowers the wave resistancebut increases the lateral force and yaw moment.

Figure 1. Illustration of the 2-D cross-section considered

Table 1. Hydrodynamic forces and yaw moment comparisonsfor deep-water (Fr = 0.267)

Sources α Cw

Cy

Cγ

Expt (Namimatsuet. al, 1985) 0º 0.0001251 0.000000 0.000000

Maruo and Song 0º 0.0001207 0.000000 0.000000(Linear wave) 10º 0.0001098 0.008365 0.003016

Present results 0º 0.0001331 0.000000 0.000000(Linear wave) 10º 0.0001238 0.008407 0.003069

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Figures 5-6 and Table 2 show the effect of finite depth onwave pattern, hydrodynamic forces and yaw moment. Thefinite depth effect is much stronger when h/d < 2 withsignificant increases in the hydrodynamic forces and yawmoment. For h/d > 5, the deep-water results are essentiallyreached.

Conclusion

The slender body assumption is invoked to reduce the non-linear flow-field around a slender ship into a series of 2-Dsections that are solved using the boundary element method.Iteration is used to handle the non-linearity at the free watersurface. Numerical results are presented for a Wigley hullwith good agreement with previous fully 3-D results. Thisindicates that the use of the slender body assumption insimplifying the flow-field computations gave acceptableengineering results and can be extended for predictions infinite depth.

References

[1] Chen X. B. and Nguyen T. (2000). Ship-Motion GreenFunction in Finite-Depth Water, Theory and Applications,Y.Goda, M. Ikehata and K. Suzuki (eds.), 2000,Yokohama, Japan, pp 151-156.

[2] Kim Y. H. and Lucas T. (1990). Non-linear Ship Waves,Proceedings of 18th Symposium on NavalHydrodynamics, pp 439-448.

[3] Maruo H. and Song W. S. (1990). Numerical Appraisalof the New Slender Ship Formulation in Steady Motion,Proceedings of 18th Symposium on NavalHydrodynamics, pp 239-257.

[4] Namimatsu M., Ogiwara S., Tanaka H., Hinatsu M. andKajitani H. (1985). An Evaluation of ResistanceComponents on Wigley Geosim Models-an Analysis andApplication of Hull Surface Pressure Measurement,Journal of Kansai Society Of Naval Architects, Japan,No.197.

[5] Shin H. R. and Takanori H. (2000). Unstructured GridFlow Solver for Ship Flows, Theory and Applications,Y. Goda, M. Ikehata and K. Suzuki (eds.), 2000,Yokohama, Japan, pp 243-248.

Figure 2. Wave profiles alongside the hull (α = 0º, Fr = 0.32)

Figure 3. Wave profiles alongside the hull (α = 10º, Fr = 0.267)

Table 2. Hydrodynamic forces and yaw moment comparisons for deep and shallow water (α = 10º, Fr = 0.267)

Wave h/d Cw

Cy

Cγ

Linear wave12 0.0001238 0.008405 0.0030681.2 0.0001498 0.010172 0.003713

Non-linear wave12 0.0001162 0.008206 0.0029131.2 0.0001408 0.009929 0.003526

Figure 4. Wave profiles alongside the hull (α = 10º, Fr = 0.267)

Figure 5. Wave profiles (non-linear wave) alongside the hull(α = 10º, Fr = 0.32)

Figure 6. Hydrodynamic forces and yaw moment coefficients(α = 10º, Fr = 0.267)

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Introduction

A computational capability for predicting ship manoeuvres isbeing developed under a current joint MPA-NTU project. Thegoal is to enhance navigational safety for ships in coastal portwaters. The predictive system is based on solving the rigidbody equations of motion for the ship, taking into account theeffects of prevailing environmental forces such as from ambientcurrents and wind, as well as the forces from the ship propulsionand rudder systems.

Modelling of ship motions

As only the ship track is desired, only the surge, sway and yawmotions (i.e. a 3-dof system) are simulated. As a first approach,empirical formulas for deep water are used to calculate theforces from the ship’s propeller and rudder, ambient wind andcurrent, and resistance from wave-making. The standardequations of surge (u), sway (v) and yaw (r) motion in a shipcentred coordinate system are:

(1)

(2)

(3)

where m is the ship mass and I is the moment of inertia. Mij

and Iij are the added masses and added moments of inertia. Theterm xG denotes the displacement of the centre of gravity fromthe ship’s centroid, which is used as the origin of the ship-centred coordinate system. The forces simulated are denotedby Fp, Fr, Fc, and Fw representing propeller, rudder, current (i.e.hull induced including wave-making) and wind forcesrespectively. The definition of the coordinate system is sketchedin Fig. 1 along with the heading angle ψ, speed V, drift angleβ and ambient current in a fixed inertial frame xo and yo.

Wind force

The wind forces are formulated as

(4)

(5)

(6)

where Vw is relative wind velocity, and Ax and Ay are theprojected cross sectional areas above the water line in the

Towards a Predictive Systemfor Ship Manoeuvres

Li Wei (cwli@ntu.edu.sg) Edmond Lo Yat Man (cymlo@ntu.edu.sg)

Tan Soon Keat (ctansk@ntu.edu.sg)

Figure 1. Definition of coordinate system

longitudinal and transverse directions respectively. The forcecoefficients Cx, Cy and CN depend on the relative wind directionfrom the ship’s bow and can be found from Harvald [1] asreproduced in Fig. 2.

Calculation of current interaction

The current force arises due to the motion of the ship relativeto the ambient current in the area. In the standard ship resistanceformulation, the total resistance R is given:

R = RW + (1+k) RF (7)

where RF is frictional resistance, RW is wave-making resistanceand k is a form factor accounting for the form drag arisingfrom the ship’s wake.

Figure 2. Wind force coefficients for ships

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For our application, we take the current Ve relative to the shipas the ambient current velocity relative to the ship velocity.Then frictional resistance along the longitudinal direction isassumed to be given by:

RF = 0.5CF Sρ|Vex|Vex (8)

where Vex is x component of Ve, S is the wetted area and CF isfrictional resistance coefficient according to ITTC 1978 frictionformulation:

(9)

with Rn being the Reynolds number based on the ship length.

The wave-making resistance is similarly taken as:

(10)

Standard values of CR as given by the ship’s Froude number Fnand prismatic coefficient can be found in Harvald [1].

The lateral current force and moment is formulated using striptheory and Vey, the y-velocity component of Ve. Essentially theship is divided into strips with each strip being treated as a 2-D section. The transverse force is written as an integral overthe sections:

(11)

where D(x) is draught of each ship section and CD(x) is the 2-D drag coefficient. Similarly the yawing moment about theship’s centroid is expressed as:

(12)

The second term in Equation (12) is the Munk moment and isneglected here.

Calculation of propeller force

The thrust T of standard propellers could be written as (see e.g.Carlton [2]):

T = Kt ρ n2D4 (13)

where Kt is the thrust coefficient, D is the propeller diameterand n is the revolution rate (Hz).

In general, Kt is given as a function of the advance ratio J:

J = Va/(Dn) (14)

where Va is the ship advance speed (Vex here) modified by awake fraction Wf to account for the interference of the hull, i.e.

Va= Vex Wf (15)

A form suggested for Wf (reference [3]) is given through the

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block coefficient Cb:

Wf = 1-0.6Cb (16)

Finally the thrust deduction fraction t further modified the finalpropeller force FP delivered as:

FP = (1-t) T (17)

Calculation of rudder force

The force on rudder normal to it surface is:

(18)

where Ar is the rudder and δr is the relative rudder angle (i.e.the attack angle relative to the flow). From this, the rudderforces and moment can be written as:

Frx = FN sinδ (19)

Fry = - FN cosδ (20)

Nr = - xr FN cosδ (21)

where xr is the location of the rudder. The force coefficient CN

can be estimated using thin airfoil theory with corrections fora finite vertical span (Harvald [1]).

Preliminary calculations

The equations of motion (Equation (1) to (3)) are solved intime using a mid-point integration scheme with coding in VisualBasic. The ship mass properties (i.e. mass and moments ofinertia) are based on a detailed calculation accounting for themass and load distribution of the ship. The added mass andmoments are estimated using potential flow for an ellipsoid inan infinite fluid domain.

A 45o course change calculation was carried out for a container-sized ship with an approach speed of 22.3 knots and the resultsare shown in Fig. 4. There are over- and under-shoots in theheading angle due to the lack of hydrodynamic damping.

A turning circle manoeuvre was also calculated at the sameapproach speed as shown in Fig. 5, where the predictions are

Figure 4. Course changing calculation

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Figure 5. Turning circle: prediction and sea trail

due to the ship forward motion need to be incorporated. Studiesare now underway using inviscid hydrodynamic calculations tobetter determine the ship wave making forces/moment and aswell as for the added mass and moments of inertia.Computational fluid dynamic calculations are also being carriedout for a double-hull ship geometry to better determine theviscous forces/moment. These calculations are also beingperformed for water of finite depth where the ship draft canoccupy a significant fraction of the water depth. However it isnoted that, eventually, a detailed comparison with ship sea traildata will be necessary in order to calibrate and verify theaccuracy of the predictions. The forcing coefficients beingassembled from various sub-theories will invariably need to befine tuned for the specific ships simulated.

Acknowledgements

The technical assistance of Det Norske Veritas, Republic ofSingapore Navy, Maritime & Port Authority of Singapore andNeptune Ship Management Services Pte Ltd in this work isgratefully acknowledged.

References

[1] Harvald, Sv. Aa., (1983), “Resistance and Propulsion ofShips”, Wiley, New York.

[2] Carlton, J. S., (1994), “Marine Propellers and Propulsionof Ships”, Oxford, Butterworth-Heinemann.

[3] Gerr, D., (1989), “Propeller Handbook”, London: Nautical.

A Diffusive Model for Evaluating Thicknessof Bedload Layer

N S Cheng (cnscheng@ntu.edu.sg)

further compared with available ship trial data. As can be seen,qualitative agreement was obtained.

Summary

The preliminary calculations indicate that the 3-dof systemcoupled with the engineering expressions for the forces andmoments could qualitatively describe the ship motion. Howevervarious refinements are needed to improve the predictions.Damping forces arising from the inviscid hydrodynamic flow

Introduction

The thickness of the bedload layer can be defined as thesaltation height of particles, which is of the order of theparticle size. This thickness is usually required to formulatethe bedload transport rate analytically. Einstein (1950) simplytook the thickness to be two particle diameters, independentof flow conditions. His assumption is acceptable only onqualitative grounds. In comparison, several subsequent studiesshow, either experimentally or numerically, that the thicknessis closely related to characteristics of flow and particles.

It seems that no analytical results are available in the literaturefor determining the thickness of the bedload layer. This paperattempts to present a framework for an analysis that can beperformed using a diffusive model. This study was partiallymotivated by recent studies on the hydrodynamic diffusion

conducted largely in chemical engineering, which indicatethat the hydrodynamic particle-particle interaction caneffectively be described as a diffusive process.Depending on flow conditions and properties of particles,different diffusion mechanisms can be identified associatedwith the transport of particles in fluids. For colloidal particles,the diffusion caused by the thermal effect is significant,leading to the Brownian motion of small particles. If theambient turbulence is intensive, the diffusive characteristicsof particles are largely associated with turbulent eddies. Forcertain conditions, the turbulent diffusion can be analogisedto the momentum transfer of fluid. It is with this assumptionthat the well-known Rouse equation is derived for computingthe concentration profile of suspended particles in openchannel flows. The hydrodynamic diffusion refers to thephenomenon that a particle executes a random walk solelybecause of the particle-particle interaction. It is not due to

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the thermal effect and can take place even at very smallparticle Reynolds numbers, implying that it is not caused bythe fluid inertia either. This phenomenon has been confirmedexperimentally and numerically.

Bedload comprises near-bed particles, which experiencecontinuous contacts with the bed. These contacts can be inthe mode of rolling, sliding or saltation. This implies thathydrodynamic interactions among the particles have acommanding influence on the bedload transport. In this study,the turbulence effect on the bedload transport is notconsidered. A diffusive model is therefore proposed fordescribing the particle flux perpendicular to the flowdirection, which enables evaluation of the thickness of thebedload layer.

Theory

Consider a bedload layer. The initial elevation of the bedsurface is zo. For a certain flow condition, an equilibriuminterface at the elevation, zc, can be identified between thebedload of which the particles are moving and the stationaryparticles beneath the bedload. The thickness of the bedloadlayer, δ, is therefore equal to the distance from the top of thebedload to the interface, i.e., (zb - zc). If the bed particlesmove typically in the mode of saltation, then the bedloadthickness can be easily measured as the saltation height.

The bed particles, once saltating, will fall down to the bedbecause of the gravitational effect. The downward flux ofthe particles, Fs, can be related to the settling velocity, wm,and the local concentration of the particles, c:

Fs = wmc (1)

where the subscript m indicates the settling velocity occurringin the particle-fluid mixture. This settling velocity can befurther expressed in terms of the settling velocity, wo, undera very dilute condition,

wm = wrwo (2)

where wr = relative settling velocity depending on theconcentration as well as the properties of the particle. Usually,it takes the following form:

wr = (1 – c)n (3)

where n can be empirically related to the dimensionlessparticle diameter, D*, or the settling Reynolds number.Alternatively, a generalized approach for evaluating wr isused in this study, as detailed in the subsequent section.

In addition to the gravity-driven particle flux, the upwardhydrodynamic diffusive flux is

(4)

where E = shear-induced diffusion coefficient. It is thisdiffusive flux that causes the bed dilatation in the verticaldirection. At equilibrium, the net flux of the particle is zero.

Equating (1) and (4) yields

(5)

Note that the particle concentration for bedload is zero abovez = zb, and approaches the maximum value, cm, for thecondition of the densely parked particles below z = zc.Integrating (5) from z = zc to z = zb for the bedload layeryields

(6)

Therefore, the elevation difference, (zb-zc), i.e., the thicknessof the bedload layer, δ, can be expressed as

(7)

From (7), it follows that the thickness can be analyticallyevaluated provided that detailed information on the diffusivity,E, and the relative settling velocity, wr, is available.

Hydrodynamic Diffusivity, E

Previous studies show that the hydrodynamic diffusivity inshear flows, E, is generally proportional to the bulk velocitygradient and the square of the particle diameter:

(8)

where dum/dy = velocity gradient of the bulk flow; um =velocity of the mixture; and E* = dimensionless diffusivitydepending on the particle concentration. The E*-valuededuced from experimental measurements generally increasewith increasing particle concentration. However, a generalformulation of such a relationship is not yet available.

Alternatively, the diffusivity is assumed in this study to beequivalent to that associated with the momentum transferinduced by the random particle motion. This implies that thediffusivity can be related to the particle-related shear stress,τp:

(9)

where ρp = density of particles. The assumption made hereis quite similar to that used for deriving the Rouse equation,where only the turbulent diffusion is concerned.

The particle shear stress, in general, may comprise threecomponents contributed by the presence of particles, randommotion of particles, and inter-particle collision, respectively.When particles are simply present in a fluid, even withoutrelative motion among them, the rheological property of thefluid is modified. On the other hand, the random motion andcollision effect lead to a ‘turbulent’ component of the particleshear stress. If the ‘turbulent’ component is predominant,the particle shear stress can be derived as

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(10)

where G = average gap among neighbouring particles and α= coefficient. Substituting (10) into (9) yields

(11)

Comparing (8) to (11) leads to

(12)

The average gap among neighbouring particles can bedetermined using the following approach. Consider a certainamount of particles dispersed randomly in a fluid, whichyields a volumetric concentration, c. On average, it is assumedthat a single particle can be enclosed in a finite volume, V,so that the ratio of the particle volume, πD3/4, to V is equalto the concentration, i.e.,

(13)

With (13), if the volume V is spherical, then its diameter isfound to be Dc−1/3, which can be considered as the averagedistance between the centres of two neighbouring particles.Therefore, the average gap, G, is equal to

(14)

Substituting (14) into (12) leads to

(15)

With G given by (14) and some existing experimental data,one can find that the α-value varies from 0.06 to 0.19.Furthermore, note that

(16)

where τb = bed shear stress; µm = µrµ = effective viscosityof the mixture; and µr = relative viscosity. Substituting (16)into (11), after manipulation, yields

(17)

Relative Viscosity, µr

Many empirical formulas are available in the literature forcomputing the relative viscosity. Their predictions differ inparticular for high concentrations. These formulas could berepresented well by the following exponential function:

(18)

where β = constant. By comparing (18) with the existingempirical relationships, the β-value is found to be equal toapproximately 1.0 to 3.9.

Relative Settling Velocity, wr

For the particle-fluid mixture, the settling Reynolds numbercan be defined as

(19)

where ρm = ρpc + ρ(1 - c) = (1 + c∆)ρ = density of themixture; and ∆ = ρp/ρ - 1. If the particle concentration isvanishingly small, Rm reduces to Ro (= ρwoD/µ). With Rm

and Ro, the relative settling velocity, wr, can be expressed as

(20)

Generally, the settling Reynolds number, Ro, is a function ofthe dimensionless particle diameter, D*. For natural sedimentparticles, this function can be formulated as:

(21)

As an approximation, (21) can be further applied to theparticle-fluid mixture, yielding:

(22)

where D*m is associated with the properties of the mixture,

(23)

Eq. (23) can be further rewritten in the following form sothat D*m can be related to D*:

(24)

With (21), (22) and (24), (20) can be rewritten as

(25)

From (18) and (25), it follows that the relative settlingvelocity varies with the concentration, specific gravity, anddimensionless diameter of particles.

Relative Thickness of Bedload Layer, δ/D

Substituting (17) into (7), one can express the relativethickness as

(26)

where E*, µr, and wr can be evaluated using (15), (18) and(25), respectively.

To the writer’s knowledge, very limited experimental dataof the thickness of the bedload layer are available in theliterature to fully verify the present study. This may bebecause the photographic technique used for tracing theparticle movement applies only to very low rates of sedimenttransport, which imply few particles saltating over a flat

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bed. In particular, a flat bed comprised of fixed particles isoften adopted in experiments. This bed is suitable forconducting measurements but not realistic. On the other hand,with increasing bed shear stress, the sediment bed with movableparticles can easily develop from the flat bed into bedformssuch as ripples and dunes. In the presence of bedforms, theheight of ripples or dunes is more measurable and meaningfulthan the saltating height of individual particles for sedimenttransport, even though they may be closely connected.

However, it is noted that at high shear stresses, bedformsmay disappear, the bed particles moving largely in the sheetmodes. For this extreme condition, the linear relationship,δ/D ~ τ*, seems to be possible, as reported by Wilson (1987)and Sumer et al. (1996). The relationship approximated firstby Wilson is δ/D = 10τ*, which agrees acceptably with Sumeret al.’s experimental data.

Conclusion

Applying the concept of the hydrodynamic diffusion, whichis caused by particle-particle interactions, to bedload transport

leads to an analytical model that enables the evaluation ofthe thickness of the bedload layer. The derived results indicatethat the bedload thickness is related to the dimensionlessparticle diameter and dimensionless bed shear stress.However, further research needs to be done to fully verifyand improve the analysis presented in this study.

References

1. Einstein, H. A. (1950). “The bed-load function forsediment transportation in open channel flow.” UnitedStates Department of Agriculture, Washington, D. C.,Technical Bulletin No. 1026.

2. Sumer, B. M., Kozakiewicz, A., Fredsoe, J., andDeigaard, R. (1996). “Velocity and concentration profilesin sheet-flow layer of movable bed.” Journal of HydraulicEngineering, ASCE, 122(10), 549-558.

3. Wilson, K. C. (1987). “Analysis of bed-load motion athigh shear stress.” Journal of Hydraulic Engineering,ASCE, 113(1), 97-103.

Total Sediment Discharge in Alluvial ChannelsLim Siow Yong (csylim@ntu.edu.sg)

Yang Shu-Qing

Introduction

The mechanism of sediment transport has been a subject ofstudy for decades due to its importance, especially in thedesign and operation of canal systems and river regulation.To date, there are many formulas for the calculation of therate of sediment discharge in alluvial channels, but so farfew have gained general acceptance. There are basically threetypes, i.e., bed-load, suspended load and total load formulae.The wash load is usually not included in these formulas. Thetotal load can be computed indirectly as the sum of the bed-load and suspended load. This method, however, contradictswhat is observed under natural flowing conditions whereboth modes of sediment motion are inter-changeable, i.e., asbed-load on one instance and as suspended load on the next,depending on the sediment characteristics and flowconditions. The purpose of this paper is to develop anempirical equation to compute the total sediment dischargebased on a new total-load transport parameter, TT and to testthe new equation using 3500 data sets covering a wide rangeof laboratory and field measurements.

Dimensional considerations

In the study of sediment transport capacity of a flow, the

following characteristic parameters are considered important:(1) flow conditions: flow depth h, channel width B, meanflow velocity V, energy slope S, and gravitational accelerationg, (2) fluid properties: specific weight γ, and kinematicviscosity of water ν, and (3) sediment properties: mediangrain diameter d50, specific weight of sediment γs, and itssubmerged weight γs - γ, and grain fall velocity ω. In sedimenttransport, the shear velocity is more meaningful than themean velocity. We will adopt the effective grain shearvelocity, proposed by van Rijn (1984) to replace V. Thisis physically more logical because for mobile bed channelswith bedforms, the grain shear is responsible for transportingthe bed material and the form drag does not contribute tothis process. The grain shear velocity can be computed using:

(1)

Therefore, the total sediment discharge gt, in dry weight persecond per unit width (N/s/m) for a steady and uniform flowin an alluvial channel can be expressed as:

(2)

Using dimensional analysis, the dimensionless form of Eq.

u’*

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(2) is:

(3)

In Eq. (3), the term B/h reflects the influence of the channelaspect ratio and needs to be considered if the wall effects aresignificant, i.e., if B/h < 5. The second term on the right sideof Eq. (3) is the Froude number based on . The term h/ν is the Reynolds number which reflects the influenceof ν. For fully developed channel flows, this term can beignored. The term h/d50 reflects the influence of the relativeroughness on the bed, and would be important if it becomes

too small. The terms, and need to be included

to reflect the specific weight of the sand used and it is moremeaningful when lightweight material is used in theexperiments or if air is used as the transporting medium.

The last term reflects the importance of the grain fall

velocity in the study of sediment transport.

Data analysis

An initial assessment of the influence of the dimensionlessparameters in Eq. (3) on gt has been conducted using Stein’s(1965) data. The 56 data sets include all stages of bed formdevelopment.

Figure 1 shows the relationships of against ,

S and S and the correlation coefficients, R obtained

were 0.66, 0.95 and 0.98, respectively. The latter R valueindicates that the sediment discharge has the best correlation

with the product, S and the trend in Figure 1 shows a

linear relationship of the form:

(4)

or (5)

Figure 1. Correlation between and S, S

based on Stein’s data (1965)

where τo = γhS = bed shear stress. Eq. (5) can be rewrittenas:

(6)

where b = constant. Substituting the initial condition forsediment transport, i.e., gt =0, when u*

’ = u*c, where u*c =critical shear velocity from the Shields curve, into Eq. (6)yields:

(7)

We seek to find an equation that can be used for both sandtransports in the field conditions as well as in laboratory. Forthe latter, lightweight materials are often used as the modelingmaterial to represent sand or gravel in the prototype. Theparameter c1 can be obtained based on the data conducted atthe U.S. Waterway Experiment Station (1936). These dataare unique because the experiments were conducted withdifferent sediment density ranging from 1030 kg/m3 to 1850kg/m3.

Figure 2 shows the variation of c1 with . The figure

shows that c1 increases almost linearly as the bed material

becomes lighter, i.e. as increases. For most

practical purposes, c1 can be expressed as:

(8)

where k = constant = 12.5. Substituting Eqs. (8) into (7), weget a total sediment discharge formula of the form:

(9)

where is defined as the total-

load transport parameter. The variables in TT can be obtainedeasily from field or flume measurement. The fall velocity inTT can be calculated using the formula proposed by Cheng(1997).

Figure 3 shows comparisons between the calculated andmeasured total load for the US Waterway ExperimentalStation data. The agreement is reasonably good with most of

Figure 2. Relationship between c1 and based on

US Waterway Experimental Station’s data (1936)

u’*

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Figure 3. Effect of sediment density on the predictability of (9),based on US Waterway Experiment Station’s data (1936).

Figure 4. Measured sediment discharge gt versus parameter TT

based on data from various sources

Table 1. Comparisons of various predictive formulas for lightweight materials using theUS Waterway Experiment Station data (1936)

ρs d50

% Score of predicted sediment discharge and range of discrepancy r

(T/m3)Runs

(mm) 0.75 < r < 1.5 0.5 < r < 2 0.33 < r < 3

Y-L* E-H VR K Y-L E-H VR K Y-L E-H VR K

1.85 26 0.96 50% 50% 61% 15% 69% 76% 85% 23% 92% 88% 92% 65%

1.85 32 0.833 44% 50% 40% 10% 69% 78% 68% 25% 81% 84% 81% 56%

1.74 29 0.833 59% 52% 62% 14% 83% 83% 75% 27% 86% 93% 89% 62%

0.97,1.35 31 3.107 35% 55% 38% 16% 71% 77% 58% 32% 87% 93% 83% 71%

1.32-1.32 64 3.0 42% 37% 36% 9% 78% 66% 62% 26% 89% 87% 84% 47%

1.26 14 1.16-4 57% 21% 7% 14% 79% 28% 14% 14% 86% 50% 71% 50%

1.29-1.11 30 2.4 60% 10% 50% 3% 73% 33% 83% 10% 93% 50% 93% 27%

0.84-1.05 72 3.2 40% 11% 22% 22% 55% 25% 55% 35% 89% 42% 86% 37%

Total 298 48% 36% 40% 13% 72% 58% 63% 24% 88% 73% 85% 52%

*Note on formulas: Y-L = Yang and Lim, E-H = Engelund and Hansen, VR = van Rijn, K = Karim

the data within the ±100% error band. Other data were alsoused to verify the applicability of Eq. (9). Altogether 3500published total load data from field and flume studies havebeen analyzed. Figure 4 shows a typical comparison betweenthe calculated and measured total load with data from varioussources. Generally, the results show that 84% of the 3500data were predicted within the 0.5 and 2 times of themeasured values, and k (= 12.5) remains a constantirrespective of the hydraulic conditions.

A comparison on the predictability of Eq. (9) with a fewformulae (Engelund and Hanson 1972, van Rijn 1984, andKarim 1998) for lightweight materials is given in Table 1,

where r = discrepancy ratio = gt (measured)/gt (computed).The last row in Table 1 shows the overall results and it canbe seen that Eq. (9) yields the best results. van Rijn’s equationalso score well for this type of sediment.

Conclusion

A new and user-friendly total sediment discharge formula(Eq. 9) for the computation of total load under equilibriumtransport condition has been developed based on dimensionalanalysis. The total sediment discharge, gt is linearly relatedto the new total-load transport parameter, TT and a factor of

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proportionality, k. Using 3500 sets of published total-loaddata from both field and flume studies, k has been found tobe a constant = 12.5. The range of the variables tested were:median sediment size from 0.011mm to 28mm, sedimentgradation up to 2.07, specific gravity of sediment from 1.03to 2.65, flow Froude number up to 2.8, sediment concentrationup to 110 kg/m3, water temperature from 1.6°C to 63°C, waterdepth from 0.03 m to 16.4 m, and channel aspect ratio assmall as 0.3. The verification exercise for the new equationshow, on average, 84% of the data were predicted within the0.5 and 2 times of the measured values. Considering the largedatabase used and the range of applicability of the formula,the result obtained is comparable, if not better than most ofthe existing total sediment discharge formulae.

References

[1] Cheng, N.S. (1997) “Simplified Settling VelocityFormula for Sediment Particle” J. Hydr. Engrg, ASCE,

123(20), 149-152.

[2] Engelund F. and Hansen, E., (1972) “A Monograph onSediment Transport in Alluvial Streams” Teknisk Forlag,Copenhagen, Denmark.

[3] Karim, F. (1998) “Bed Material Discharge Predictionfor Nonuniform Bed Sediment” J. Hydr. Engrg., ASCE,124(6), 597-603.

[4] Stein, R.A. (1965) “Laboratory studies of total load andapparent bed load” J. of Geophysical Research, Vol. 70,No.8, 1831-1842.

[5] van Rijn, L.C. (1984) “Sediment transport Part I: bedload transport” J. Hydr. Engrg. ASCE, 110(10), 1431-1456.

[6] U.S. Waterways Experiment Station (1936) “Studies oflight-weight materials with special reference to theirmovement and use as model bed material” United StatesArmy Corps of Engineers. Technical Memorandum 103-1.

Double Diffusive Effect on Mixed BrineDischarges from Desalination

A W K Law (cwklaw@ntu.edu.sg)

Introduction

A concern with brine discharges from desalination plants isthe increase in salinity in the surrounding waters near thesea bed level due to poor initial mixing. The inducedstratification can inhibit the exchange of matter between thebenthic layer and the overlying ambient waters, and lead toan accumulation of chemicals from the pre-treatment processin the bottom sediment. Thus, the brine discharge issometimes merged with neighboring cooling water beforebeing released through outfalls. Besides the dilution withthe merging, the mixture of heat and salinity is alsodeliberately arranged to yield an almost neutrally buoyantdischarge (i.e. the density of the discharge being the same asthe ambient waters), so that the mixed brine plume wouldnot have the tendency to sink and hence can be pushedfurther out to the sea. A mixed discharge of heat and salinityis also the natural outcome of desalination with a multi-stage distillation approach.

To assess the initial mixing of the mixed discharges, thepossibility of differential transport of heat and salinity throughdouble diffusion needs to be addressed. The much largermolecular diffusivity of heat than salt can lead to thephenomenon of salt-fingering in the case of an overlayingwarm salty layer. This drives a downward salinity flux asa result of their relatively cooler and saltier anomalies thanthe surrounding ambient. With salt-fingering, the trajectoryof a mixed jet can differ significantly from what is intendedto be a neutrally buoyant jet.

In this study, we investigate the effect of double diffusion onthe mixed brine discharge through laboratory experiments.The study is divided into two parts. The first is with asurface rectangular jet that emulates the prototype geometry,while a submerged round jet is studied in the second part.For both, the discharge has the same density as the ambientwater, with the increase in density due to the excess salinitycompensated by the equivalent reduction due to the elevatedtemperature. Without initial density differences, classicalanalysis would suggest that the jet would simply propagatehorizontally. Due to space constraint here, we shall onlydiscuss the results and analysis of the submerged round jet.

Experiments

The experiments were conducted in a glass test tank withdimensions 2.86 m length x 0.86 m width x 1 m depth. Themixed brine was issued from a horizontal circular pipe withan inside diameter D of 5 mm and located at approximatelymid-depth. The densities of both the ambient and dischargedfluid were monitored by a digital density probe. Fluorescentdye was used to track the path of salt transport.

12 tests were conducted that can be grouped into threedifferent sets, TA, TB and TC, with varying discharge velocityUo of 1.6, 0.4 and 0.08 m/s respectively. Sets TA and TBwere in the turbulence range, while the flow in Set TC wasactually laminar due to the low Uo. Controlled experiments

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were performed without any temperature rise for the firsttest in each of the three sets. For all the controlledexperiments, the jet propagated horizontally towards the endwall as anticipated.

In the first Set TA, the trajectories were also near horizontal upto approximately 60D for the three tests with ∆T at 4, 8 and12oC respectively. Thereafter the jet showed a downwardinclination of up to 3D towards the end of the monitoring range.

Set B saw similar actions with the jet gaining negativebuoyancy due to the accumulation of buoyancy anomalyand curving downwards. Sample snapshots for Set TB areshown in Figure 1, and the trajectories in Figure 2. The jettrajectory for Tests TB3 and TB4, with a ∆T of 8 and 12oCrespectively, are quite similar. Test TB2 with ∆T= 4oC hasa smaller curvature comparatively. A further decrease of Uo

(a) ∆T = 4oC

(b) ∆T = 8oC

(c) ∆T = 12oC

Figure 1. Snapshots of the submerged round jet in Set TB

in Set TC showed even more significant double-diffusiveeffect.

Analysis

In this section, we shall perform a quantitative analysis ofthe trajectory of a round jet that is subjected to doublediffusion action. The analysis begins with the commoncharacteristic of a buoyant jet, that the horizontal momentumflux Mx remains constant along the trajectory.

(1)

where u is the axial velocity and θ the jet inclination. Thevertical momentum, on the other hand, is accelerated by thedownward buoyancy as:

(2)

where b the jet width and the average density excess overthe cross-section.

When salt fingers develop over the bottom interface, excessdensity accumulates inside the fingers due to the salinitysurplus over the surrounding ambient. The excess density isthe source of the negative buoyancy that forces the downwardmigration of the salinity. The rate of increase of the totaldensity excess inside the fingers should thus be proportionalto the salinity flux across the interface. Hence, we canexpress the cross-sectional average density buildup, / ρa , as follows:

(3)

(4)

where is the average salinity and Q the volume flux.From Wang and Law (2002), the increase in the dischargeand jet width of a round jet with distance can be quantifiedas:

(5)

(6)

with kjw = 6.48, η jw = 0.105.

In the presence of shear turbulence, the change in salinityinside the jet due to turbulent entrainment is large comparedto the salt-fingering flux. The average salinity can then becomputed as:

(7)

Substituting the above equations into (2) and assuming thatthe jet is near-horizontal yield

(8)

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where

(9)

The trajectory of the jet can now be estimated as:

(10)

Therefore

(11)

The power law in Equation (11) shows a high exponent of11/3 for the horizontal distance, which explains why the jetcan bend downward rapidly despite having the same densityas the ambient initially. The relatively longer trajectory forTest TB2 was used to calibrate the value of γ, yielding γ =5.14. The calculated trajectories for Set TB are shown inFig 2 respectively.

A few observations can be drawn here. First, the overallagreement between the calculated and measured trajectoriesis satisfactory given the uncertainties. Second, the relativeeffect on the trajectory due to the salinity increase over thethree experiments in each individual set is well captured,implying the validity of (4) in correlating the excess densitywith the cross-sectional average salinity. Third, the coefficientγ remains unchanged throughout the range of dischargevelocity. Since Set TA is well into the turbulent range, themagnitude of γ should also be applicable in prototypeconditions.

Figure 2. Trajectories for Set TB

Summary

The present study shows that the double diffusive salt-fingering action can lead to a strong migration of the brinesalinity towards the seabed, and should be taken into accountin the environmental assessment. Analysis shows that thedownward salinity migration is triggered by the excess densitybuildup inside the salt-fingering interface. Further verificationof the results in the field scale will be necessary in thefuture.

Reference

[1] Wang, H.W., and Law, A.W.K., 2002. “Second orderintegral model of a round buoyant jet.” J. FluidMech., Vol.459, pp397-428.