e> Pergamon

PU: 50273-1223(98)00767-7

Wat. Sci. T~ch. Vol.38. No.10.pp. 331-336.1998.IAWQ

Il:I1998 Published by ElsevierScienceLtd.Printed in Oreal Britain. All rightsreserved

0273-1223198519'00 +0'00

DILUTION IN A DENSE BOTTOM JET INCROSS CURRENTS

Ole Petersen*'** and Torben Larsen*

• DepartmentofCivilEngineering. AalborgUniversity, Sohngaardsholmsvej 57,DK·9000.Denmark•• International Research Centrefor Computational Hydrodynamics, Agem Alli 5.DK·2970 Hersholm; Denmark

ABSTRACf

A )-dimensional numerical model describing the dilution in the near field around dense vertical jets in across flow is fonnulated and validated against laboratory experiments. The validation shows that the modelreproduces the flow pattern well. though the dilution is underestimated by 20%. The model is applied 10 acase study where the dilution from two vertical jets at an angle in shallow water is described. It isdemonstrated that a 20% increase in dilution is possible. II is concluded that the model may become avaluable 1001 in diffusor design. @ 1998 IAWQ Published by Elsevier Science Ltd. All rights reserved

KEYWORDS

Sea outfalls; initial dilution; dense waste water; industrial waste water.

INTRODUCTION

Discharge of wastewater through diffuser ports located at the seabed is a very common solution to disposefresh waste water, thus numerous studies exist in the literature on the dilution of buoyant jets and plumesunder a variety of conditions. Disposal of heavier negatively buoyant wastewater is more unusual, thus theknowledge of the dilution in the near field is here more sparse and no standard design methods areimmediately available.

Previous studies on initial dilution of heavy jets have mainly been concerned with disposal of brine.Stolzenbach and Adams, (1980) (SA), developed an integral model for the dilution in heavy jets while Hollyand Grace, (1972), made a series of model experiments. Based on an experimental study Petersen and Larsen(1994) developed a model for dilution in plumes with small density differences in currents. Related workshave been presented by Baines et al. (1990) and Chen et al. (1991). The numerical modelling of jets andplumes has more generally been discussed by (Larsen, 1994). In the present paper the initial dilution inheavy vertical jets in a cross current is discussed by means of a 3-D numerical model that can describe thecomplex flow pattern that arises around such a jet.

Below we present the numerical model of the flow in the near field. the model is verified using previouslaboratory experiments and the model is finally used in the discussion of the design of an actual diffuserarrangement.

3)1

332 O. PElERSEN andT. LARSEN

THE NUMERICAL MODEL

The numerical model is based on the 3-dimensional Reynolds equation, where buoyancy effects aredescribed using a transport equation for salinity, and is coupled to the momentum equation usingBoussinesq's hypothesis and a linear equation of state. The turbulent mixing is described using a standard k-emodel, that incorporates the effects of both stable and unstable stratifications on the turbulence (Rodi, 1980).The resulting 7 coupled partial differential equations are solved on a staggered cartesian grid using aniterative method that retains all couplings between the equations. The coupling between continuity andmomentum equations are made using a so-called Simple method. The discretisations are made using a Pecletnumber dependent upwinding scheme. The equations are solved using the model system PHOENICS(Rosten and Spalding, 1985).

BOUNDARYCONDnJONS

The model is set up to a situation with a steady current approaching one side of the computational domain,as sketched in Figure I. Due to the symmetry around a plane through the diffuser and parallel to the current,only one half of the domain is actually modelled. The surface is a rigid lid and a log wall law is used alongthe bed. The diffuser ports are modelled simply as a volume discharge and a momentum discharge throughone cell face at the bed. From the general experience with momentum jets it is believed that only the field inthe immediate neighbourhood of the port is affected by this relatively crude approximation, while the flowfurther away is controlled mainly by the momentum source. Outlet boundary conditions (upwinddifferencing and no diffusive transport) is used at the side away from the diffuser and the downstream side.This enables fluid to leave the domain freely.

~l,/ Qo

" Do" ~

Figure t. Sketchof a vertical densejet in a crosscurrent.

RESULTS

Dense bottom jets in a cross current

When the volume discharge in the dense bottom jet is relatively small, it can be argued that the flow anddilution will depend on a densimetric Froude number FD defined from port conditions as

(1)

where Va is the ambient velocity, the density, Do is the diameter of the discharge port, dp the densitydifference at the port and g the gravitational acceleration. The jet initially rises, as shown in Figure I, due tothe momentum in the discharge, while it entrains ambient water and is deflected by the cross current.Gradually the negative buoyancy takes over and the jet bends downward where it impacts on the bed. Acomplicated 3 dimensional flow may arise here with fluid spreading out laterally with internally supercritical flow and an upstream intrusion of the denser water.

Dilution in a densebonomjet in crosscurrents 333

A characteristic dynamical length scale for the rise height can be given as 12m = QoUc/lfla. Experiments byStolzenbach and Adams (1980) with relatively low FD's show that characteristic scales at the maximumheight zMcan be estimated as zM=1M FDo the reattachment length XI =2.81MFD2 and the dilution SasS oc

Qo/ (4UaI2mF4'30) '

Numerical results

The numerical model is verified by a comparison with the experimental correlations summarized inStolzenbach and Adams (1980). Four combinations of discharges and currents, covering FD from 0.2 to 0.7as shown in Table I, are used as the basis for the verification. The reduced gravity g ~p/p at the port is in allcases 0.20, the discharge is 0.047 m3/s and the port diameters are 0.087 m and 0.12 m, respectively. Thewater depth is 10 m.

Table I. Calculated dilutions at point of impact

# U. Uo FD S.(mls) (m1s) 0 0

1.00 0.04 4.00 0.26 40.00

2.00 0.04 8.00 0.31 133.00

3.00 0.08 4.00 0.52 60.00

4.00 0.08 8.00 0.61 200.00

The domain is resolved using 40 x 20 x 40 points with a 1.05 rate of increase in spacing away from thediffusor ports. This resolution gives a gridsize independent solution. The limited size of the domain and theapproximate boundary conditions at the outlet boundaries give some reflection of internal waves, butcalculations with larger domains show that the dilution at the point of impact is not significantly affected.

Figure2. The +0.1kg/m3 isodensity surfaceas calculated by themodel . Thesituation is Ua. 0.04mis,Uo.. 8 mil, Qo =: 0.047m3/s• .1p .. +20kg/m3, corresponding to run1/2.

334 O. PETERSEN and T. LARSEN

In Figure 2 is shown one example of the concentration field around the jet port in terms of a +0.1 kglm 3 iso­density surface. It appears that the entrainment in the rising jet is reproduced as the jet expands along theaxis. In this situation the jet actually reaches the surface, where it spreads out due to the residual momentum.Hereafter the negative buoyancy drives the jet downwards again where it impacts the bottom. For higherambient currents the bend over of the jet will be larger while for more buoyant jets the discharge eventuallywill act as a fountain (Baines et ai., 1990). Also some influence from the outlet boundaries is seen.

_ ~ ~_..i _ ~ L ' ;....;. : .. ; : ;.: : .

S, , .;." ,

1 ~~~~~:~~~~~~~~~~~~1~~~~~~~~~~;~~J~~~~[~~~~~~~~~~~~!~!~~!!~l!~f:••••.••••••_••..•.__•.••+..•••.••.•..; ~...••...':"•.••.~ f-.

······_·······_··········~···········_-;···_·····i········:-····+·_··:···-+···i··_ _ ·····_··-4···.-·······-i _ ~ t ••••••+•••••l._.+ ~..

: i i : : i

F;,

Figure3. Experimental correlation for dilutionat the pointof impingement (Stolzenbach and Adams,1980)andcalculated values,shownas function of Fo-

The dilution at the point of impact is here estimated as ratio between the maximum density just above theimpact area I and the density at the diffusor port S = t.plt.PI' In Figure 3 is shown modelled andexperimentally esti- mated dilutions as summarized in Stolzenbach and Adams, 1990. It appears that thenumerical model, almost consistently, underestimates the measured dilutions by 20%. This may be acombined effect of a coarse resolution near the diffusor port and differences in measuring techniques. Thereis also a good agreement for other characteristics such as maximum height and reattachment length xI' It isconcluded that the agreement is satisfactory.

CASE STUDY

Danish coastal waters may roughly be characterized by alongshore tidal currents at 0.1 to 0.3 mls and waterdepths of the order 10m. In such shallow water special care must be given to the design of the diffusorarrangement in order to achieve a sufficient initial dilution. Therefore the model has been applied to aspecific study where the feasibility of using an outlet arrangement with two diffuser ports is studied. Theoutlet arrangement is shown in Figure 4, where the total discharge is divided between two diffusor ports,placed with a I m spacing in the cross main current direction with an outward angle at 30°.

I \ I \~ /;/.~ ....

Figure 4. Diffuserports.

Dilution ina dense bottom jetincross currents 33S

Experience with wall jets and multiport diffusors shows that due to the entrainment a low pressure zone willdevelop between the jets that tends to merge the two jets with reduced entrainment and dilution as aconsequence. This is here counteracted by the angling of jets. In Figure Sa and Sb is shown an example ofone calculation with Uo =8 mls and Ua =0.04 mls. The conclusions from several runs were that the presentarrangement may give up to a 20 % increase in dilution.

Fig. Sa

Fig.Sb.

Figure S. (a)The +0.15 kg/m3isodensity surface ascalculated bythe model for two angled diffusor ports; (b)isodensity curves along bed. The situation isUal: 0.08 mis, Uol: 4 mis, Qo l: 0.023 m3/s per port, fop .. +20 kg/m 3.

336

DISCUSSION

O. PETERSEN and T. LARSEN

Although emphasis here has been on the dilution at the point of impact it appears that the model doesreproduce the most important characteristics in the flow around the diffusor port and gives a fair descriptionof the dilution. The internal hydraulic jump that surrounds the' point of inception is reproduced as is theupstream penetration of dense water that in some conditions may form a horse shoe shaped sharp front(Stolzenbach and Adams, 1980). However, further work is needed before firm quantitative predictions canbe made. A similar model has been successfully verified for the flow in the dense, expanding bottom plumethat under some conditions may form a distance downstream of the diffusor port (petersen and Larsen,1994).

The general conclusion for this study is that 3 dimensional numerical models may become a valuable tool instudies of dilution around diffusor ports as they can incorporate site specific deviations from experimentallyderived design criteria. The more specific conclusions can be summarized as:

A 3D numerical model that reproduces the experimentally measured dilution in vertical dense jets in crosscurrents has been set up.The model shows that it may be feasible to use two diffuser ports in order to improve initial dilution inshallow water.

REFERENCES

Baines, W. D., Turner, J. S. and Cambell, I. H. (1990). Turbulent fountains in an open chamber. J. Fluid Mech., %12, 557-592.Chen, H. B, Larsen, T. and Petersen, O. (1991). Turbulent Buoyant Jets in Flowing Ambients. In: Environmental Hydraulics, I,

97-102, Lee, J. H. W. and Cheung, Y. K. (eds), Balkema, Rotterdam.Holly, F. M. and Grace. J. L. (1972). Model studies of dense jets in flowing fluid. J. HydrauL Diy., ASCE, 98, (HYll).Larsen. T. (1994). Numerical modelling of jets and plumes - a civil engineering perspective. In: Recent Research Advances in the

Fluid Mechanics of Turbulent Jets and Plumes. Davies. P. A. and M. J. Valente Neve. (eds), 223-235. Kluwer AcademicPublishers, The Netherlands.

Petersen, O. and Larsen, T. (1994). Dilution of dense bottom plumes in turbulent currents. In: Proc. 4th Int. Symp. on StratifiedFlows, 3, Hopfinger, E., Voisin, B. and Genevieve, C. (eds), Grenoble, France.

Petersen, O. (1994). Laboratory and numerical experiments on the dilution of buoyant surface plumes. In: Recent ResearchAdvances in the Fluid Mechanics of Turbulent Jets and Plumes. Davies, P. A. and M. J. Valente Neves (eds), 223-235.Kluwer Academic Publishers. the Netherlands.

Rodi, W. (1980). Turbulence Models and Their Applications in Hydraulics. IAHR State of the art review, Delft, The Netherlands.Rosten, H. I. and Spalding, D. B. (1985). Phoenics User Manual. CHAM Ltd, London.Stolzenbach, K. D. and Adams, E. E. (1980). Submerged discharge of dense effluent In: Stratified flows, Carstens, T. and

McClimans, T. (eds), Trondheim, Norway. Tapir. 832-844.