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TITLE: Technical Guidance Manual for Performing Wasteload Allocations, 

Book III: Estuaries – 

Part 3: Use of Mixing Zone Models in Estuarine Waste Load Allocations 

EPA DOCUMENT NUMBER: EPA-823-R92-004 DATE: August 1992 

POINT OF CONTACT: TBD 

ABSTRACT 

As part of ongoing efforts to keep EPA’s technical guidance readily accessible to 

water quality practitioners, selected publications on Water Quality Modeling and 

TMDL Guidance available at http://www.epa.gov/waterscience/pc/watqual.html 

have been enhanced for easier access. 

This document is part of a series of manuals that provides technical information 

related to the preparation of technically sound wasteload allocations (WLAs) that 

ensure that acceptable water quality conditions are achieved to support 

designated beneficial uses. The document provides information on the legal 

requirements for mixing zones and describes the background and application of 

predictive models for mixing zone analysis. 

Book III Part 3 describes how the detailed prediction of conditions in the initial 

mixing phase of a wastewater discharge relates to legal definitions of mixing 

zones. It also gives an overview of the major physical processes that govern 

hydrodynamic mixing of aqueous discharges, and introduces (1) jet integral 

models that are applicable to a limited subset of near-field processes and (2) the 

CORMIX expert system for mixing zones, that addresses both near-field and farfield processes under a variety of conditions. Four case studies are included to 

illustrate the application of the jet integral models and of CORMIX. 

KEYWORDS: Wasteload Allocations, Estuaries, Modeling, Water Quality 

Criteria, Mixing Zones 
United States

Environmental Protection

Agency

Office of Water

(WH 585)

EPA-823-R-92-004

August 1992

TechnicalGuidanceManual

forPerformingWasteLoad

Allocations

Book III:Estuaries

Part 3

Use Of Mixing Zone Models

In Estuarine Waste Load

Allocations
TECHNICAL GUIDANCE MANUAL

FOR PERFORMING WASTE LOAD ALLOCATIONS

BOOK III: ESTUARIES

PART 3: Use of Mixing Zone Models in Estuarine Waste Load

Allocations

Project Officer

Hiranmay Biswas, Ph.D.

Edited By

Robert B. Ambrose, Jr., P.E.1

James L. Martin, Ph.D., P.E.2

Prepared by

Gerhard H. Jirka, Ph.D., P.E.3

1. Center for Exposure Assessment Modeling,

Environmental Research Laboratory, U.S. EPA, Athens, GA

2. ASCI Corporation, U.S. EPA, Athens, Georgia

3. DeFrees Hydraulics Laboratory, School of Civil and Environmental Engineering,

Cornell University, Ithaca, NY

Prepared for

U.S. ENVIRONMENTAL PROTECTION AGENCY

401 M. Street, S.W.

Washington, D.C. 20460

Table of Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..xiii

7. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.1 Initial Mixing in Estuaries and Coastal Waters . . . . . . . . . . . . . . . . . 7-1

7.2 Mixing Zone Requirements: Legal Background . . . . . . . . . . . . . . . . 7-1

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

7.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

8. Physical Processes and Modeling Methodologies . . . . . . . . . . . . . . . . . . 8-1

8.1 Ambient and Discharge Conditions . . . . . . . . . . . . . . . . . . . . . . 8-1

8.2 Hydrodynamic Mixing Processes . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.3 Mathematical Predictive Models . . . . . . . . . . . . . . . . . . . . . . . . 8-5

8.4 Buoyant Jet Integral Models . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7

8.5 CORMIX: Expert System Methodology for Mixing Zone Analysis . . . . . . 8-11

8.6 Mixing Zone Predictions Under Unsteady Reversing Tidal Currents . . . . 8-23

8.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-25

9. Case Studies of Mixing Zone Prediction . . . . . . . . . . . . . . . . . . . . . . . 9-1

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

9.2 Case AA - Single Port Discharge: Industrial Outfall in Tidal Fjord . . . . . . . 9-2

9.3 Case BB - Multiport Diffuser: Municipal Sewage Discharge

into Coastal Bay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5

9.4 Case CC - Single Port Discharge: Brine Discharge From

an Oil Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8

9.5 Case DD Multiport Diffusers: Cooling Water Discharge

into Shallow Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-10

9.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-12

Glossary

Acute Toxicity1 - Any toxic effect that is produced

within a short period of time, usually 24-96 hours.

Although the effect most frequently considered is

mortality, the end result of acute toxicity is not necessarily death. Any harmful biological effect may be the

result.

Aerobic1 - Refers to life or processes occurring only

in the presence of free oxygen; refers to a condition

characterized by an excess of free oxygen in the

aquatic environment.

Algae (Alga)1 - Simple plants, many microscopic,

containing chlorophyll. Algae form the base of the

food chain in aquatic environments. Some species

may create a nuisance when environmental conditions are suitable for prolific growth.

Allochthonous1

- Pertaining to those substances, materials or organisms in a waterway which originate

outside and are brought into the waterway.

Anaerobic2 - Refers to life or processes occurring in

the absence of free oxygen; refers to conditions characterized by the absence of free oxygen.

Autochthonous1 - Pertaining to those substances,

materials, or organisms originating within a particular

waterway and remaining in that waterway.

Autotrophic1 - Self nourishing; denoting those organisms that do not require an external source of organic

material but can utilize light energy and manufacture

their own food from inorganic materials; e.g., green

plants, pigmented flagellates.

Bacteria1

- Microscopic, single-celled or noncellular

plants, usually saprophytic or parasitic.

Benthal Deposit2 - Accumulation on the bed of a

watercourse of deposits containing organic matter

arising from natural erosion or discharges of wastewaters.

Benthic Region1 - The bottom of a waterway; the

substratum that supports the benthos.

Benthal Demand2 - The demand on dissolved oxygen

of water overlying benthal deposits that results from

the upward diffusion of decomposition products of the

deposits.

Benthos1 - Organisms growing on or associated principally with the bottom of waterways. These include:

(1) sessile animals such as sponges, barnacles, mussels, oysters, worms, and attached algae; (2) creeping forms such as snails, worms, and insects; (3)

burrowing forms, which include clams, worms, and

some insects; and (4) fish whose habits are more

closely associated with the benthic region than other

zones; e.g., flounders.

Biochemical Oxygen Demand2 - A measure of the

quantity of oxygen utilized in the biochemical oxidation of organic matter in a specified time and at a

specific temperature. It is not related to the oxygen

requirements in chemical combustion, being determined entirely by the availability of the material as a

biological food and by the amount of oxygen utilized

by the microorganisms during oxidation. Abbreviated

BOD.

Biological Magnification1 - The ability of certain organisms to remove from the environment and store in

their tissues substances present at nontoxic levels in

the surrounding water. The concentration of these

substances becomes greater each higher step in the

food chain.

Bloom1 - A readily visible concentrated growth or

aggregation of minute organisms, usually algae, in

bodies of water.

Brackish Waters1 - Those areas where there is a

mixture of fresh and salt water; or, the salt content is

greater than fresh water but less than sea water; or,

the salt content is greater than in sea water.

Channel Roughness2 - That roughness of a channel,

including the extra roughness due to local expansion

or contraction and obstacles, as well as the roughness

of the stream bed proper; that is, friction offered to the

flow by the surface of the bed of the channel in contact

with the water. It is expressed as roughness coefficient in the velocity formulas.

Chlorophyll1 - Green photosynthetic pigment present

in many plant and some bacterial cells. There are

seven known types of chlorophyll; their presence and

abundance vary from one group of photosynthetic

organisms to another.

Chronic Toxicity1 - Toxicity, marked by a long duration, that produces an adverse effect on organisms.

The end result of chronic toxicity can be death although the usual effects are sublethal; e.g., inhibits

reproduction, reduces growth, etc. These effects are

reflected by changes in the productivity and population structure of the community.

v
Coastal Waters1 - Those waters surrounding the continent which exert a measurable influence on uses of

the land and on its ecology. The Great Lakes and the

waters to the edge of the continental shelf.

Component Tide2 - Each of the simple tides into

which the tide of nature is resolved. There are five

principal components; principal lunar, principal solar,

N2, K, and O. There are between 20 and 30 components which are used in accurate predictions of tides.

Coriolis Effect2

- The deflection force of the earth’s

rotation. Moving bodies are deflected to the right in

the northern hemisphere and to the left in the southern

hemisphere.

Datum2 - An agreed standard point or plane of state

elevation, noted by permanent bench marks on some

solid immovable structure, from which elevations are

measured or to which they are referred.

Density Current2 - A flow of water through a larger

body of water, retaining its unmixed identity because

of a difference in density.

Deoxygenation2 - The depletion of the dissolved oxygen in a liquid either under natural conditions associated with the biochemical oxidation of organic matter

present or by addition of chemical reducing agents.

Diagenetic Reaction - Chemical and physical

changes that alter the characteristics of bottom sediments. Examples of chemical reactions include oxidation of organic materials while compaction is an

example of a physical change.

Dispersion2 - (1) Scattering and mixing. (2) The mixing of polluted fluids with a large volume of water in a

stream or other body of water.

Dissolved Oxygen2 - The oxygen dissolved in water,

wastewater, or other liquid, usually expressed in milligrams per liter, or percent of saturation. Abbreviated

DO.

Diurnal2 - (1) Occurring during a 24-hr period; diurnal

variation. (2) Occurring during the day time (as opposed to night time). (3) In tidal hydraulics, having a

period or cycle of approximately one tidal day.

Drought2 - In general, an extended period of dry

weather, or a period of deficient rainfall that may

extend over an indefinite number of days, without any

quantitative standard by which to determine the degree of deficiency needed to constitute a drought.

Qualitatively, it may be defined by its effects as a dry

period sufficient in length and severity to cause at

least partial crop failure or impair the ability to meet a

normal water demand.

Ebb Tide1

- That period of tide between a high water

and the succeeding low water; falling tide.

Enrichment1 - An increase in the quantity of nutrients

available to aquatic organisms for their growth.

Epilimnion1 - The water mass extending from the

surface to the thermocline in a stratified body of water;

the epilimnion is less dense that the lower waters and

is wind-circulated and essentially homothermous.

Estuary1 - That portion of a coastal stream influenced

by the tide of the body of water into which it flows; a

bay, at the mouth of a river, where the tide meets the

river current; an area where fresh and marine water

mix.

Euphotic Zone1 - The lighted region of a body of water

that extends vertically from the water surface to the

depth at which photosynthesis fails to occur because

of insufficient light penetration.

Eutrophication1 - The natural process of the maturing

(aging) of a lake; the process of enrichment with

nutrients, especially nitrogen and phosphorus, leading to increased production of organic matter.

Firth1 - A narrow arm of the sea; also the opening of

a river into the sea.

Fjord (Fiord)1 - A narrow arm of the sea between

highlands.

Food Chain1 - Dependence of a series of organisms,

one upon the other, for food. The chain begins with

plants and ends with the largest carnivores.

Flood Tide2 - A term indiscriminately used for rising

tide or landward current. Technically, flood refers to

current. The use of the terms “ebb” and “flood” to

include the vertical movement (tide) leads to uncertainty. The terms should be applied only to the horizontal movement (current).

Froude’s Number2 - A numerical quantity used as an

index to characterize the type of flow in a hydraulic

structure that has the force of gravity (as the only force

producing motion) acting in conjunction with the resisting force of inertia. It is equal to the square of

characteristic velocity (the mean, surface, or maximum velocity) of the system, divided by the product

of a characteristic linear dimension, such as diameter

or expressed in consistent units so that the combinations will be dimensionaless. The number is used in

vi
open-channel flow studies or in cases in which the free

surface plays an essential role in influencing motion.

Heavy Metals2 - Metals that can be precipitated by

hydrogen sulfide in acid solution, for example, lead,

silver, gold, mercury, bismuth, copper.

Heterotrophic1 - Pertaining to organisms that are

dependent on organic material for food.

Hydraulic Radius2 - The right cross-sectional area of

a stream of water divided by the length of that part of

its periphery in contact with its containing conduit; the

ratio of area to wetted perimeter. Also called hydraulic

mean depth.

Hydrodynamics2 - The study of the motion of, and the

forces acting on, fluids.

Hydrographic Survey2 - An instrumental survey

made to measure and record physical characteristics

of streams and other bodies of water within an area,

including such things as location, areal extent and

depth, positions and locations of high-water marks,

and locations and depths of wells.

Inlet1 - A short, narrow waterway connecting a bay,

lagoon, or similar body of water with a large parent

body of water; an arm of the sea, or other body of

water, that is long compared to its width, and that may

extend a considerable distance inland.

Inorganic Matter2 - Mineral-type compounds that are

generally non-volatile, not combustible, and not biodegradable. Most inorganic-type compounds, or reactions, are ionic in nature, and therefore, rapid

reactions are characteristic.

Lagoon1 - A shallow sound, pond, or channel near or

communicating with a larger body of water.

Limiting Factor1 - A factor whose absence, or excessive concentration, exerts some restraining influence

upon a population through incompatibility with species requirements or tolerance.

Manning Formula2 - A formula for open-channel flow,

published by Manning in 1890, which gives the value

of c in the Chezy formula.

Manning Roughness Coefficient2 - The roughness

coefficient in the Manning formula for determination

of the discharge coefficient in the Chezy formula.

Marsh1 - Periodically wet or continually flooded area

with the surface not deeply submerged. Covered

dominantly with emersed aquatic plants; e.g., sedges,

cattails, rushes.

Mean Sea Level2 - The mean plane about which the

tide oscillates; the average height of the sea for all

stages of the tide.

Michaelis-Menton Equation2 - A mathematical expression to describe an enzyme-catalyzed biological

reaction in which the products of a reaction are described as a function of the reactants.

Mineralization2 - The process by which elements

combined in organic form in living or dead organisms

are eventually reconverted into inorganic forms to be

made available for a fresh cycle of plant growth. The

mineralization of organic compounds occurs through

combustion and through metabolism by living animals. Microorganisms are ubiquitous, possess extremely high growth rates and have the ability to

degrade all naturally occurring organic compounds.

Modeling2 - The simulation of some physical or abstract phenomenon or system with another system

believed to obey the same physical laws or abstract

rules of logic, in order to predict the behavior of the

former (main system) by experimenting with latter

(analogous system).

Monitoring2 - Routine observation, sampling and testing of designated locations or parameters to determine efficiency of treatment or compliance with

standards or requirements.

Mouth2 - The exit or point of discharge of a stream into

another stream or a lake, or the sea.

Nautical Mile2 - A unit of distance used in ocean

navigation. The United States nautical mile is defined

as equal to one-sixteenth of a degree of a great circle

on a sphere with a surface equal to the surface of the

earth. Its value, computed for the Clarke spheroid of

1866, is 1,853.248 m (6,080.20ft). The International

nautical mile is 1,852 m (6,070.10 ft).

Nanoplankton2 - Very minute plankton not retained in

a plankton net equipped with no. 25 silk bolting cloth

(mesh, 0.03 to 0.04 mm.).

Neap Tides1 - Exceptionally low tides which occur

twice each month when the earth, sun and moon are

at right angles to each other; these usually occur

during the moon’s first and third quarters.

Neuston2 - Organisms associated with, or dependent

upon, the surface film (air-water) interface of bodies

of water.

Nitrogenous Oxygen Demand (NOD)2 - A quantitative measure of the amount of oxygen required for the

biological oxidation of nitrogenous material, such as

vii
ammonia nitrogen and organic nitrogen, in wastewater; usually measured after the carbonaceous oxygen

demand has been satisfied.

Nutrients1 - Elements, or compounds, essential as

raw materials for organism growth and development;

e.g., carbon, oxygen, nitrogen, phosphorus, etc.

Organic1 - Refers to volatile, combustible, and sometimes biodegradable chemical compounds containing

carbon atoms (carbonaceous) bonded together and

with other elements. The principal groups of organic

substances found in wastewater are proteins, carbohydrates, and fats and oils.

Oxygen Deficit1 - The difference between observed

oxygen concentration and the amount that would

theoretically be present at 100% saturation for existing conditions of temperature and pressure.

Pathogen1 - An organism or virus that causes a disease.

Periphyton (Aufwuchs)1 - Attached microscopic organisms growing on the bottom, or other submersed

substrates, in a waterway.

Photosynthesis1 - The metabolic process by which

simple sugars are manufactured from carbon dioxide

and water by plant cells using light as an energy

source.

Phytoplankton1 - Plankton consisting of plant life.

Unattached microscopic plants subject to movement

by wave or current action.

Plankton1 - Suspended microorganisms that have

relatively low powers of locomotion, or that drift in the

water subject to the action of waves and currents.

Quality2 - A term to describe the composite chemical,

physical, and biological characteristics of a water with

respect to it’s suitability for a particular use.

Reaeration2 - The absorption of oxygen into water

under conditions of oxygen deficiency.

Respiration1 - The complex series of chemical and

physical reactions in all living organisms by which the

energy and nutrients in foods is made available for

use. Oxygen is used and carbon dioxide released

during this process.

Roughness Coefficient2 - A factor, in the Chezy,

Darcy-Weisbach, Hazen-Williams, Kutter, Manning,

and other formulas for computing the average velocity

of flow of water in a conduit or channel, which represents the effect of roughness of the confining material

on the energy losses in the flowing water.

Seiche1 - Periodic oscillations in the water level of a

lake or other landlocked body of water due to unequal

atmospheric pressure, wind, or other cause, which

sets the surface in motion. These oscillations take

place when a temporary local depression or elevation

of the water level occurs.

Semidiurnal2 - Having a period or cycle of approximately one half of a tidal day. The predominating type

of tide throughout the world is semidiurnal, with two

high waters and two low waters each tidal day.

Slack Water2 - In tidal waters, the state of a tidal

current when its velocity is at a minimum, especially

the moment when a reversing current changes direction and its velocity is zero. Also, the entire period of

low velocity near the time of the turning of the current

when it is too weak to be of any practical importance

in navigation. The relation of the time of slack water

to the tidal phases varies in different localities. In

some cases slack water occurs near the times of high

and low water, while in other localities the slack water

may occur midway between high and low water.

Spring Tide1 - Exceptionally high tide which occurs

twice per lunar month when there is a new or full

moon, and the earth, sun, and moon are in a straight

line.

Stratification (Density Stratification)1 -Arrangement of water masses into separate, distinct, horizontal layers as a result of differences in density; may be

caused by differences in temperature, dissolved or

suspended solids.

Tidal Flat1 - The sea bottom, usually wide, flat, muddy

and nonproductive, which is exposed at low tide. A

marshy or muddy area that is covered and uncovered

by the rise and fall of the tide.

Tidal Prism2 - (1) The volume of water contained in a

tidal basin between the elevations of high and low

water. (2) The total amount of water that flows into a

tidal basin or estuary and out again with movement of

the tide, excluding any fresh-water flows.

Tidal Range2 - The difference in elevation between

high and low tide at any point or locality.

Tidal Zone (Eulittoral Zone, Intertidal Zone)1 - The

area of shore between the limits of water level fluctuation; the area between the levels of high and low tides.

Tide1 - The alternate rising and falling of water levels,

twice in each lunar day, due to gravitational attraction

viii
of the moon and sun in conjunction with the earth’s

rotational force.

Tide Gage2 - (1) A staff gage that indicates the height

of the tide. (2) An instrument that automatically registers the rise and fall of the tide. In some instruments,

the registration is accomplished by printing the heights

at regular intervals; in others by a continuous graph in

which the height of the tide is represented by ordinates

of the curve and the corresponding time by the abscissae.

Toxicant1 - A substance that through its chemical or

physical action kills, injures, or impairs an organism;

any environmental factor which, when altered, produces a harmful biological effect.

Water Pollution1 - Alteration of the aquatic environment in such a way as to interfere with a designated

beneficial use.

Water Quality Criteria1 - A scientific requirement on

which a decision or judgement may be based concerning the suitability of water quality to support a designated use.

Water Quality Standard1 - A plan that is established

by governmental authority as a program for water

pollution prevention and abatement.

Zooplankton2 - Plankton consisting of animal life.

Unattached microscopic animals having minimal capability for locomotion.

1Rogers, B.G., Ingram, W.T., Pearl, E.H., Welter, L.W.

(Editors). 1981, Glossary, Water and Wastewater

Control Engineering, Third Edition, American Public

Health Association, American Society of Civil Engineers, American Water Works Association, Water

Pollution Control Federation.

2

Matthews, J.E., 1972, Glossary of Aquatic Ecological

Terms, Manpower Development Branch, Air and

Water Programs Division, EPA, Oklahoma.

ix

Preface

The document is the third of a series of manuals

providing information and guidance for the preparation

of waste load allocations. The first documents provided

general guidance for performing waste load allocations

(Book I), as well as guidance specifically directed

toward streams and rivers (Book II). This document

provides technical information and guidance for the

preparation of waste load allocations in estuaries. The

document is divided into four parts:

Part 1 of this document provides technical information

and policy guidance for the preparation of estuarine

waste load allocations. It summaries the important

water quality problems, estuarine characterisitics and

processes affecting those problems, and the simulation models available for addressing these problems.

Part 2 provides a guide to monitoring and model calibration and testing, and a case study tutorial on simulation of waste load allocation problems in simplified

estuarine systems. Part 4 summarizes several historical case studies, with critical review by noted experts.

This part, “Part 3: Use of Mixing Zone Models in

Estuarine Wasteload Allocations” describes the initial

mixing of wastewater in estuarine and coastal environments and mixing zone requirements. The important

physical processes that govern the hydrodynamic mixing of aqueous discharges are detailed, followed by

application of available models to four case study

situations.

A draft version of this document received scientific

peer review from the following modeling experts:

Donald R.F. Harleman,

Massachusetts Institute of Technology

Gerald T. Orlob,

University of California-Davis

Robert V. Thomann,

Manhattan College

Steven J. Wright,

University of Michigan

Their comments have been incorporated into the final

version.

Organization: “Technical Guidance Manual for Performing Waste Load Allocations. Book III:

Estuaries”

Part Title

1 Estuaries and Waste Load Allocation Models

2 Application of Estuarine Waste Load Allocation Models

3 Use of Mixing Zone Models in Estuarine Waste Load Allocation Modeling

4 Critical Review of Estuarine Waste Load Allocation Modeling

Acknowledgements

The contents of this section have been removed to

comply with current EPA practice.

xiii

7. Introduction

7.1 Initial Mixing in Estuaries and Coastal

Waters

The discharge of waste water into an estuary or coastal

water body can be considered from two vantage points

regarding its impact on ambient water quality.

On a larger scale, or system wide context, care must

be taken that water quality conditions that protect

designated beneficial uses are achieved. This is the

realm of the general waste load allocation (WLA) procedures and models as discussed in the first two parts

of this manual. As noted, an additional benefit of a

technically sound WLA is that excessive degrees of

treatment which are neither necessary nor productive

of corresponding improvements in water quality for the

whole water body, or at least major sections thereof,

can be avoided.

On a local scale, or in the immediate discharge vicinity,

additional precautions must be taken to insure that

high initial pollutant concentrations are minimized and

constrained to small zones, areas or volumes. The

definition of these zones, commonly referred to as

“mixing zones”, is embodied in United States water

quality regulations, on the Federal and/or State level.

The mixing zone is a legally defined spatial quantity -

with certain size and shape characteristics - that allows

for initial mixing of the discharge. More recent regulations on discharges of toxic substances define an

additional subregion - labeled herein the “toxic dilution

zone” - within the usual mixing zone. The intent of

those regulations is to require rapid mixing of toxic

releases to limit the exposure of aqueous flora and

fauna to elevated concentrations. The detailed prediction of pollutant concentrations and water quality constituents in the initial mixing phase of a wastewater

discharge is the realm of mixing zone models. This is

the subject of this part of the manual. Mixing zone

models are intended to document for any given combination of discharge and environmental conditions the

size and shape of legally defined “mixing zones”, and

for toxic substances, of embedded “toxic dilution

zones”, and the levels of pollutant concentration within

these zones and at their edge.

There may be a great diversity in the types of initial

mixing processes for wastewater discharge. First, the

size and flow characteristics of estuaries or coastal

water can vary widely: the water body may be deep or

shallow, stagnant or flowing, and may exhibit ambient

density stratification of various degrees. Secondly, the

discharge type and configuration can be highly variable: the flow may contain various pollutants ranging

from conventional to toxic substances, vary greatly in

magnitude ranging from low flowrate for a small sewage

treatment plant to the substantial cooling water flow for

a large steam-electric power plant, issue with high or low

velocity, be denser or lighter than the ambient, be located near shore or far offshore, and exhibit various

geometric details ranging from single port submerged

discharges to multiport submerged diffusers to surface

discharges.

Given this diversity of both discharge and ambient environmental conditions, there are a large number of possible flow patterns which will develop as the discharge

waste stream mixes in the ambient water. These flow

patterns will determine the configuration, size, and intensity of the mixing process, and any impact of the

discharge on the water body surface, bottom, shoreline,

or other areas. This, in turn, requires that engineering

analyses, in the form of mixing zone models, be robust,

adaptable and reliable under a wide spectrum of flow

conditions.

7.2 Mixing Zone Requirements: Legal

Background

7.2.1 Pollutant Types

The Clean Water Act of 1977 defines five general categories of pollutants: i) conventional, ii) nonconventional,

iii) toxics, iv) heat, and v) dredge and fill spoil. The Act

distinguishes between new and existing sources for

setting effluent standards. Table 7-1 lists examples of

the first three pollutant categories.

Pollutants designated as “conventional” would be “generally those pollutants that are naturally occurring, biodegradable, oxygen demanding materials and solids. In

addition, compounds which are not toxic and which are

similar in characteristics to naturally occurring, biodegradable substances are to be designated as conventional pollutants for the purposes of the provision”.

Pollutants designated as “nonconventional” would be

“those which are not toxic or conventional” (Congressional Research Service, 1978).

7.2.2 Mixing Zone Definitions

The mixing zone is defined as an “allocated impact

zone” where numeric water quality criteria can be exceeded as long as acutely toxic conditions are prevented. A mixing zone can be thought of as a limited

area or volume where the initial dilution of a discharge

occurs (USEPA, 1984a). Water quality standards apply

at the boundary of the mixing zone, not within the mixing

zone itself. USEPA and its predecessor agen7-1

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cies have published numerous documents giving guidance for determining mixing zones. Guidance published by USEPA in Water Quality Standards Handbook

(1984a) supersedes these sources.

In setting requirements for mixing zones, USEPA

(1984a) requires that “the area or volume of an individual zone or group of zones be limited to an area or

volume as small as practicable that will not interfere

with the designated uses or with the established community of aquatic life in the segment for which the uses

are designated,” and the shape be “a simple configuration that is easy to locate in the body of water and

avoids impingement on biologically important areas,”

and “shore hugging plumes should be avoided.”

The USEPA rules for mixing zones recognize the State

has discretion whether or not to adopt a mixing zone

and to specify its dimensions. USEPA allows the use

of a mixing zone in permit applications except where

one is prohibited in State regulations. A review of

individual State mixing zone policies shows that 48 out

of 50 States (the exceptions are Arizona and Pennsylvania) make use of a mixing zone in some form

(USEPA, 1984a, 1985). State regulations dealing with

streams or rivers generally limit mixing zone widths or

cross-sectional areas, and allow lengths to be determined on a case-by-case basis.

In the case of lakes, estuaries and coastal waters,

some states specify the surface area that can be

affected by the discharge. (The surface area limitation

usually includes the underlying water column and benthic area.) If no specific mixing zone dimensions are

given the actual shape and size can be determined on

a case-by-case basis.

Special mixing zone definitions have been developed

for the discharge of municipal wastewater into the

coastal ocean, as regulated under Section 301(h) of

the Clean Water Act (USEPA, 1982). For those discharges the mixing zone was labeled as the “zone of

initial dilution” in which rapid mixing of the waste stream

(usually the rising buoyant fresh water plume within the

ambient saline water) takes place. USEPA (1982) requires that the “zone of initial dilution” be a regularly

shaped area (e.g. circular or rectangular) surrounding

the discharge structure (e.g. submerged pipe or diffuser

line) that encompasses the regions of high (exceeding

standards) pollutant concentrations under design conditions. In practice, limiting boundaries defined by dimensions equal to the water depth measured horizontally

from any point of the discharge structure are accepted

by the USEPA provided they do not violate other mixing

zone restrictions (USEPA, 1982).

7.2.3 Special Mixing Zone Requirements for

Toxic Substances

USEPA maintains two water quality criteria for the allowable concentration of toxic substances: a criterion maximum concentration (CMC) to protect against acute or

lethal effects; and a criterion continuous concentration

(CCC) to protect against chronic effects (USEPA, 1985).

The less restrictive criterion, the CCC, must be met at

the edge of the same regulatory mixing zone specified

for conventional and nonconventional discharges.

In order to prevent lethal concentrations of toxics in the

regulatory mixing zone, the restrictive CMC criterion

must be met within a short distance from the outfall or

in the pipe itself. If dilution of the toxic discharge in the

ambient environment is allowed, this requirement, which

will be defined here as a toxic dilution zone (TDZ), is

usually more restrictive than the legal mixing zone for

conventional and nonconventional pollutants. USEPA

(1991) recommends four alternatives for preventing

acute lethality. One alternative is to require that the CMC

be achieved within the pipe itself. The other three alternatives allow the use of a TDZ.

The first of these involves a high-velocity discharge

combined with a mixing zone spatial limitation. For this

option, USEPA recommends a minimum exit velocity of

3 meters per second (10 feet per second) and a spatial

limitation of 50 times the discharge length scale in any

direction. The discharge length scale is defined as the

square root of the cross-sectional area of any discharge

outlet.

The next alternative recommended by USEPA (1991) is

not to use a high-velocity discharge, but rather to ensure

that the most restrictive of the following conditions is

met:

• The CMC must be met within 10% of the distance

from the edge of the outfall structure to the edge of

the regulatory mixing zone in any spatial direction.

Conventional Nonconventional Toxic

biochemical oxygen

demand (BOD)

chemical oxygen

demand (COD)

chloroform/lead

pH fluoride fluorene

total suspended solids (TSS)aluminum nickel

fecal coliform bacteriasulfide selenium

oils and grease ammonia benzidine

Table 7-1. Examples of Conventional, Nonconventional, and

Toxic Pollutants [USEPA 1984 b]

7-2

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• The CMC must be met within a distance of 50

times the discharge length scale in any spatial

direction. The discharge length scale is defined as

the square-root of the cross-sectional area of any

discharge outlet. This restriction is intended to

ensure a dilution factor of at least 10 within this

distance under all possible circumstances, including situations of severe bottom interaction and

surface interaction.

• The CMC must be met within a distance of 5 times

the local water depth in any horizontal direction.

The local water depth is defined as the natural

water depth (existing prior to the installation of the

discharge outlet) prevailing under mixing zone

design condition (e.g. low flow for rivers). This

restriction will prevent locating the discharge in

very shallow environments or very close to shore,

which would result in significant surface and bottom concentrations. (USEPA, 1991)

The latter of these geometric restrictions essentially

eliminates the use of surface (canal-type) discharges

for the discharge of toxic pollutants.

The final recommended alternative is for the discharger to show that a drifting organism would not be

exposed to 1-hour average concentrations exceeding

the CMC, or would not receive harmful exposure when

evaluated by other valid toxicological analysis

(USEPA, 1991).

7.3 Summary

The following two chapters in Part 3 of this manual deal

with the background and the application of predictive

models for mixing zone analysis that address the

various legal requirements as outlined above.

Chapter 8 first gives an overview of the important

physical processes that govern the hydrodynamic mixing of aqueous discharges. Emphasis is put herein on

submerged discharges, because of the practical limitations on surface discharges, in particular as regards

toxic pollutants. Those processes are divided into

near-field processes (influenced directly by the discharge geometry and dynamics and, to some extent,

controllable through appropriate design choices) and

into far-field processes (influenced primarily by the existing environmental conditions). It is shown that legal

mixing zone requirements can encompass, in general,

processes in both near-field and far-field. Then the

mathematical background and formulations for different

mixing zone models are reviewed. For practical routine

applications, these models fall into two classes: (i) jet

integral models that are applicable only to a sub-set of

near-field processes including submerged buoyant jets

without any boundary (surface or bottom) interaction,

and (ii) a mixing zone expert system, CORMIX, that

addresses both near-field and far-field processes under

a variety of discharge and ambient conditions.

Chapter 9 illustrates the application of jet integral models

and of the expert system CORMIX. Typical data requirements for the implementation of these models are discussed. Four case studies are presented in order to

demonstrate the capabilities and/or limitations of individual models.

7.4 References

Congressional Research Service, 1978. Legislative History of the Clean Water Act 1977. Congressional Research Service, Library of Congress, October 1978, No.

95-14 P. 330.

USEPA. 1982. Revised Section 301 (h) Technical Support Document. EPA 430/9-82-011, Washington, DC.

USEPA. 1984a. Water Quality Standards Handbook.

Office of Water Regulations and Standards, Washington, DC.

USEPA. 1984b. Technical Guidance Manual for the

Regulations Promulgated Pursuant to Section 301 (g)

of the Clean Water Act of 1977 (Draft). Washington, DC,

August.

USEPA. 1985. Technical Support Document for Water

Quality-based Toxics Control. Office of Water, Washington, DC, September.

USEPA. 1991. Technical Support Document of Water

Quality-based Toxics Control. Office of Water, Washington, DC, March.

7-3

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8. Physical Processes and Modeling Methodologies

8.1 Ambient and Discharge Conditions

The mixing behavior of any wastewater discharge is

governed by the interplay of ambient conditions in the

receiving water body and by the discharge characteristics.

The ambient conditions in an estuary or coastal water

body are described by geometric parameters - such as

plan shape of the estuary, vertical cross-sections, and

bathymetry, especially in the discharge vicinity and by

its dynamic characteristics. The latter are given by the

velocity and density distribution in the estuary, again

primarily in the discharge vicinity.

Many estuaries are highly energetic water bodies and

their velocity field with its vertical and temporal variability may be influenced by many factors. Usually the

most significant velocity component is controlled by

tidal influences, but freshwater inflows, wind-driven

currents and wave-induced currents may also play

important roles and, in some cases, may even dominate the flow. Furthermore the mean velocity field is

often superposed by secondary currents due to topographic effects or due to baroclinic influences giving

rise to complicated three-dimensional flow fields.

The density distribution in estuaries is usually strongly

coupled with the velocity field. Density differences are

mostly caused by the freshwater inflow and lighter,

less saline, water tends to overflow the more saline

ocean water. Estuaries are sometimes classified on

the basis of their density structure into well-stratified,

partially-stratified and vertically mixed estuaries (Fischer et al., 1979). Well stratified estuaries, usually

those with weak tidal effects, exhibit a two-layer structure with an upper predominantly fresh water layer

flowing over a lower saline layer (the so-called salt

wedge). The dominant vertical velocity distribution in

that instance is a seaward flow in the upper layer and

a reversed landward flow in the lower layer. The other

end of the spectrum is given by vertically well mixed

estuaries with strong tidal energetics leading to nearly

complete vertical mixing although density gradients

may still exist in the horizontal direction (i.e. along the

axis of the estuary or tidal bay).

Clearly, a major feature of estuarine ambient conditions is their time variability. For tidally controlled currents this is given by a time scale equal to the tidal

period. Other time scales, usually also of the order of

several hours, describe wind driven currents and

seiche motions. However, the time scale for initial

mixing processes of effluent discharges is usually much

shorter (of the order of minutes to tens of minutes) so

that it usually suffices to analyse certain flow and density

conditions under a steady-state assumption. The consideration of tidal reversals and potential pollutant accumulation is discussed further below (Section 8.6).

The discharge conditions relate to the geometric and

flux characteristics of the submerged outfall installation.

For a single port discharge the port diameter, its elevation above the bottom and its orientation provide the

geometry; for multiport diffuser installations the arrangement of the individual ports along the diffuser line,

the orientation of the diffuser line and construction

details represent additional geometric features. The flux

characteristics are given by the discharge flow rate from

the port, by its momentum flux and by its buoyancy flux.

The buoyancy represents the relative density difference

between discharge and ambient that, upon multiplication with the gravitational acceleration, is a measure of

the tendency for the effluent flow to rise (for positive

buoyancy) or to fall (for negative buoyancy).

8.2 Hydrodynamic Mixing Processes

The hydrodynamics of an effluent continuously discharging into a receiving body of water can be conceptualized as a mixing process occurring in two separate

regions. In the first region, the initial jet characteristics

of momentum flux, buoyancy flux, and outfall geometry

influence the jet trajectory and mixing. This region will

be referred to as the “near-field”, and encompasses the

buoyant jet subsurface flow and any surface or bottom

interaction, or in the case of a stratified ambient, terminal layer interaction. In this region, designers of the

outfall can usually affect the initial mixing characteristics

through appropriate manipulation of design variables.

As the turbulent plume travels further away from the

source, the source characteristics become less important. Conditions existing in the ambient environment will

control trajectory and dilution of the turbulent plume

through buoyant spreading motions and passive diffusion due to ambient turbulence. This region will be

referred to here as the “far-field”.

It is stressed at this point that the distinction between

near-field and far-field is made purely on hydrodynamic

grounds. It is unrelated to any legal mixing zone definitions that address prescribed water quality standards as

discussed in Section 7.2.2. In many practical cases the

legal mixing zone may, in fact, include near-field hydrodynamic mixing processes. But that does not

8-1
have to be so: For example, buoyant jet mixing in a

deep environment with crossflow may extend far beyond a legal mixing zone that is defined by a horizontal

length equal to the water depth on the basis of section

301(h) of the Clean Water Act (USEPA, 1982). As a

counter-example, a small source in a strong crossflow

may rapidly enter the passive far-field diffusion region

(in form of a bottom attached plume) well before the

edge of a legal mixing zone! Thus, in principle, the

whole gamut of mixing processes, ranging from the

near-field to the far-field, must be considered for individual mixing zone analyses.

8.2.1 Near-Field Processes

8.2.1.1 Buoyant Jet Mixing

The effluent flow from a submerged discharge port

provides a velocity discontinuity between the discharged fluid and the ambient fluid causing an intense

shearing action. The shearing flow breaks rapidly

down into a turbulent motion. The width of the zone of

high turbulence intensity increases in the direction of

the flow by incorporating (“entraining”) more of the

outside, less turbulent fluid into this zone. In this manner, any internal concentrations (e.g. of fluid momentum or of pollutants) become diluted by the

entrainment of ambient water. Inversely, one can

speak of the fact that both fluid momentum and pollutants become gradually diffused into the ambient field.

The initial velocity discontinuity may arise in different

fashions. In a “pure jet” (also called “momentum jet” or

“non-buoyant jet”), the initial momentum flux in the form

of a high-velocity injection causes the turbulent mixing.

In a “pure plume,” the initial buoyancy flux leads to local

vertical accelerations which then lead to turbulent mixing. In the general case of a “buoyant jet” (also called a

“forced plume”), a combination of initial momentum flux

and buoyancy flux is responsible.

Thus, buoyant jets are characterized by a narrow turbulent fluid zone in which vigorous mixing takes place.

Furthermore, depending on discharge orientation and

the direction of buoyant acceleration, generally curved

trajectories are established in a stagnant uniform-density environment (see Figure 8-1).

Buoyant jet mixing is further affected by ambient currents and density stratification. The role of ambient

currents is to gradually deflect the buoyant jet into the

current direction and to induce additional mixing. The

role of ambient stratification is to counteract the vertical

acceleration within the buoyant jet leading ultimately

Figure 8-1. Buoyant jet with round or slot geometry in stagnant uniform-density environment.

8-2
to trapping of the flow at a certain level (trapping level

or terminal level).

8.2.1.2 Boundary Interaction Processes and Near-Field

Stability

Ambient water bodies always have vertical boundaries: these are the water surface and the bottom, but

in addition “internal boundaries” may exist in the form

of layers of rapid density change (pycnoclines). Depending on the dynamic and geometric characteristics

of the discharge flow, a variety of interaction phenomena can occur at such boundaries. Furthermore, in the

case of a continuously (e.g. linearly) stratified ambient

where flow trapping may occur, other interaction phenomena may take place.

In essence, these interaction processes provide a

transition between the buoyant jet mixing process in

the near-field, and between buoyant spreading and

passive diffusion in the far-field.

Interaction processes can be (i) gradual and mild or (ii)

abrupt leading to vigorous transition and mixing processes. (i) If a buoyant jet is bent-over by the cross-flow

it will gradually approach the surface, bottom or terminal

level and will undergo a smooth transition with little

additional mixing.

(ii) If a jet is impinging normally, or near-normally, on a

boundary, it will rapidly spread in all directions (see

Figure 8-2). Different possibilities exist at that point: (a)

If the flow has sufficient buoyancy it will ultimately form

a stable layer at the surface (Figure 8-2a,c). In the

presence of weak ambient flow this will lead to an

upstream intrusion against the ambient current. (b) If the

buoyancy of the effluent flow is weak or its momentum

very high, unstable recirculation phenomena can occur

in the discharge vicinity (see Figure 8-2b,d). This local

recirculation leads to re-entrainment of already mixed

water back into the buoyant jet region. Thus, simple

buoyant jet analyses no longer suffice to predict these

phenomena.

The aspect of near-field stability, i.e. the distinction into

stable or unstable conditions, is a key feature of pollution analyses. “Stable discharge” conditions, usually

occurring for a combination of strong buoyancy, weak

momentum and deep water, are often referred as “deep

Figure 8-2. Stable or unstable near-field flows produced by submerged buoyant discharges.

8-3
water” conditions. “Unstable discharge” conditions, on

the other had, may be considered synonymous to

“shallow water” conditions. Further detail on discharge

stability can be found in Jirka (1982 a,b) and Holley

and Jirka (1986).

Yet another type of interaction process concerns submerged buoyant jets discharging in the vicinity of the

water bottom into a stagnant or crossflowing ambient.

Two types of dynamic interaction processes can occur

that lead to rapid attachment of the effluent plume to

the water bottom (see Figure 8-3). These may be wake

attachment forced by the crossflow or Coanda attachment (due to low pressure effects) forced by the entrainment demand of the effluent jet itself. In either

case the assumption of free buoyant jets is invalidated

and other analyses have to be pursued for these

bottom-attached flows.

8.2.1.3 Multiport Diffuser Induced Flows in Shallow

Water (Intermediate-Field)

Some multiport diffuser installations represent large

sources of momentum, while their buoyancy effects

may be relatively weak. Therefore these diffusers will

have an unstable near-field with shallow water conditions. This is characteristic, for example, for cooling

water diffusers from thermal power plants. For certain

diffuser geometries (i.e. the unidirectional and the

staged diffuser types; see Section 8.3) strong motions

can be induced in the shallow water environment in the

form of vertically mixed currents that laterally entrain

ambient water and may extend over long distances

before they re-stratify or dissipate their momentum. In

a sense, these “diffuser plumes” extend beyond the

strict near-field (of the order of the water depth) and are

sometimes referred to as the “intermediate-field” (Jirka,

1982b).

8.2.2 Far-Field Processes

In the context of this report, far-field mixing processes

are characterized by the longitudinal advection of the

mixed effluent by the ambient current velocity.

8.2.2.1 Buoyant Spreading Processes

Buoyant spreading processes are defined as the horizontally transverse spreading of the mixed effluent flow

while it is being advected downstream by the ambient

current. Such spreading processes arise due to the

buoyant forces caused by the density difference of the

mixed flow relative to the ambient density. If the discharge is nonbuoyant, or weakly buoyant, and the ambient is unstratified, there is no buoyant spreading

region in the far-field, only a passive diffusion region.

Depending on the type of near-field flow and ambient

stratification several types of buoyant spreading may

occur: (i) spreading at the water surface, (ii) spreading

at the bottom, (iii) spreading at a sharp internal interface

(pycnocline) with a density jump, or (iv) spreading at the

terminal level in continuously (e.g. linearly) stratified

ambient fluid.

As an example, the definition diagram and structure of

surface buoyant spreading processes in unstratified

crossflow is shown in Figure 8-4. The laterally spreadFigure 8-3. Bottom attachment processes for submerged

discharges.

Figure 8-4. Buoyant spreading processes in the far-field

(Example: surface spreading).

8-4
ing flow behaves like a density current and entrains

some ambient fluid in the “head region” of the current.

The mixing rate is usually relatively small. Furthermore, the flow may interact with a nearby bank or

shoreline (not shown in the figure). The layer thickness

may decrease during this phase.

Depending on source and ambient characteristics,

buoyant spreading processes can be effective transport mechanisms that can quickly spread a mixed

effluent laterally over large distances in the transverse

direction. This can be particularly pronounced in cases

of strong ambient stratification in which the effluent at

the terminal level that may initially be of considerable

vertical thickness collapses into a thin but very wide

layer unless this is prevented by lateral boundaries.

8.2.2.2 Passive Ambient Diffusion Processes

The existing turbulence in the ambient environment

becomes the dominating mixing mechanism at sufficiently large distances from the discharge point. The

intensity of this passive diffusion process depends

upon the geometry of the ambient shear flow as well

as any existing stratification. In general, the passively

diffusing flow is growing in width and in thickness (see

Figure 8-5). Furthermore, it may interact with the channel bottom and/or banks.

The strength of the ambient diffusion mechanism depends on a number of factors relating mainly to the

geometry of the ambient shear flow and the ambient

stratification. In the context of classical diffusion theory

(i.e. gradient diffusion, see Fischer et al., 1979) diffusion processes in bounded flows (e.g. rivers or narrow

estuaries) can be described by constant diffusivities in

the vertical and horizontal direction that depend on

turbulent intensity and on channel depth or width as

the length scales. On the other hand, wide “unbounded” channels or open coastal areas are characterized by plume size dependent diffusivities leading to

accelerating plume growth described, for example, by

the “4/3 law” of diffusion. In the presence of a stable

ambient stratification the vertical diffusive mixing is generally strongly damped.

8.3 Mathematical Predictive Models

8.3.1 Modeling Methodology

In principle, one can conceive of two approaches to the

prediction of effluent discharges in the water environment: complete models or zone models.

(i) Complete models: These are three-dimensional numerical models that directly solve a finite difference or

finite element approximation for the full dynamic and

mass conservation equations with various assumptions

for the turbulent shear and mass transport terms. In

principle, with the advent of powerful computing facilities, even on the desktop, such a complete modeling

approach that encompasses the entire fluid domain of

interest with all individual mixing processes appears

feasible. However, successful applications to date have

been limited. Apparent reasons for the present shortcomings include (1) lack of fully workable turbulence

closure techniques under the influence of buoyancy

while considering the full range of jet-induced geophysical turbulence; (2) the difficult trade-off of modeling a

large enough domain while providing sufficient resolution in a three-dimensional model (computer capacity

and costs); and (3) the unknown nature of the open fluid

boundary conditions which need to be specified as part

of the elliptic equation system. These boundaries may,

in general, contain a combination of stratified inflow and

outflow that is inherently difficult to specify. For these

reasons, complete numerical models are usually not

used in routine mixing zone analyses of effluent discharges and this is expected to remain so for at least

the next decade.

(ii) Zone Models: Instead of attempting to integrate the

general governing equations over the whole region of

interest it is frequently useful to divide the region into

several zones with distinct behavior (such as individual

mixing processes in the near-field and in the far-field).

Within these zones it is then possible to simplify the

governing equations by dropping unimportant terms.

This gives a considerable advantage in the mathematical treatment and improved accuracy in the solution.

However, a challenge remains because the solutions

are restricted to specific zones. Thus, criteria need to

be established for a meaningful division of the whole

region into zones, and to provide transition conditions

between zones.

Figure 8-5. Passive ambient diffusion processes.

8-5
Current practice in pollution analyses relies on zone

models. Such models that deal with individual flow

processes are described in the specialized research

literature as well as in several monographs (e.g. Fischer et al., 1979, Holley and Jirka, 1986). However, a

problem arises because there is limited guidance to

the model user on the limits of applicability of each

model, and on how to combine the individual models

for an overall prediction of the entire flow process. The

use of an integrated expert system framework (see

below) promises to alleviate this problem.

An important group of zone models are the so-called

buoyant jet integral models that are limited to the

buoyant jet mixing process as described in Section

8.2.1.1 without attention to any problems of boundary

interaction and near-field instability. Several of such

integral model formulations are available as computer

programs. Whenever their applicability has been ascertained, these models have been found through

numerous data-model comparisons to be reliable and

accurate. Jet integral models will be reviewed in Section 8.4.

An integrated framework of zone models for all important near-field and far-field mixing processes that effect

effluent mixing has recently been developed. This

framework is in the form of an expert system that

classifies each discharge/ambient condition as to

which flow processes are important and provides a

prediction through a sequence of zone models with

appropriate transition conditions. The zone modeling

expert system methodology CORMIX (Doneker and

Jirka, 1990; Akar and Jirka, 1991) is discussed in

Sections 8.5 and 8.6.

8.3.2. Zone Model Schematizations of

Discharge and Ambient Conditions

All zone models require some schematization of the

complex and arbitrary ambient and discharge conditions that may prevail at any discharge site. These

simplifications are needed to conform to the requirements of the individual models.

A schematic definition diagram for a single port discharge is given in Figure 8-6. The bottom is assumed

to be flat (constant depth) while any banks (if considered in the analysis) are assumed to be vertical.

A corresponding diagram for multiport diffusers is provided in Figure 8-7. Of particular interest for this case

is the alignment angle γ between the crossflow direction and the diffuser axis, the orientation angle β between the individual port axes and the diffuser line, and

the vertical angle θ between port axis and the horizontal plane. Three major diffuser types have evolved in

actual design practice and can be characterized by

these angles (see Figure 8-8).

In the unidirectional diffuser, all the ports point in the

same direction perpendicular to the diffuser axis

( β=90o). In the staged diffuser, all ports point along the

diffuser line ( β = 0

o

). In the alternating diffuser, the

ports are arranged in an alternating fashion and point in

opposite directions ( β=± 90o). The undirectional and

the staged diffusers possess a net horizontal momentum input with a tendency to induce currents within the

ambient water body. The alternating diffuser has a zero

net horizontal momentum, and a lesser tendency to

generate currents and circulations.

Of course, there are variations on the basic theme for

each of the three diffuser types. Some of these design

possibilities are shown in Figure 8-8. There may be

double or triple nozzle arrangements (with a small internal angle) for both unidirectional or staged diffusers, and

the port orientation angle β may differ somewhat from

the nominal value, 90o or 0o, respectively. Or, in case

of the alternating diffuser, there may be multiple port

assemblies for each riser with several ports arranged in

a circular fashion. Furthermore, alternating diffusers for

thermal discharges in shallow water may have a variable port orientation along the diffuser axis to control

instabilities and horizontal circulations (for details, see

Jirka, 1982b). Another special case of

Figure 8-6. Schematics of single port discharge geometry in

ambient channel with rectangular cross-section

(width W may be finite or unlimited).

Figure 8-7. Schematics of multiport diffuser geometry in

ambient channel with rectangular cross-section

(width W may be finite or unlimited).

8-6
an alternating diffuser is given by a vertical discharge

from all ports.

Any diffuser can be deployed with arbitrary alignment

γ. However, the two major arrangements are the

perpendicular alignment ( γ ≈ 90 o) and the parallel

alignment ( γ ≈ 0o).

8.4 Buoyant Jet Integral Models

8.4.1 Basic Elements: Stagnant Unstratified

Ambient

The narrow elongated shape of the turbulent zone

within a buoyant jet (see Figure 8-1) suggests boundary-layer type simplifications to the equations of fluid

motion and mass transport. The equations may be

further simplified by integrating across the local jet

cross-section thereby yielding a one-dimensional

equation set for the actual three-dimensional problem.

This is the essence of jet integral models which solve

the equation set with a simple integration scheme

marching forward along the trajectory.

The integral method is demonstrated in the following

for a round buoyant jet issuing into a stagnant unstratified ambient (Figure 8-1). The jet-trajectory is assumed

to lie within an x-z coordinate system. Local integration

across the buoyant jet gives the following flux (integral)

quantities:

V

olume flux: Q = 2π∫u

0

∞

r d r = 2π I1 uc b

2

(1)

Momentum flux (kinematic):

M = 2 π ∫

o

∞

u

2

r d r = 2 π I2 u c

2

b

2

(2)

Scalar (pollutant) mass flux:

Q c = 2 π ∫

o

∞

u c r d r = 2π I3 uc cc b

2

 (3)

 Buoyancy flux:

J = 2π ∫

o

∞

u g ′ r d r = 2π I3 uc gc

′

b

2

(4)

in which u = mean velocity in the trajectory direction, r

= transverse coordinate from local jet centerline, c =

mean concentration, and g

′

 = mean buoyant acceleration relative to the outside fluid where

g

′

 =

ρa − ρ

ρa

g (5)

ρ = local density, ρa = ambient density, and g = gravitational acceleration. In the rightmost integrated quantities, the subscript c indicates centerline values, and the

width b is a measure of the width of the jet (see below).

The profile constants I 1, I 2, I 3, are simple numerical

values that depend on the chosen profile shape and on

the width definition (see Holley and Jirka, 1986). Frequently, a bell-shaped Gaussian profile is

Figure 8-8. Schematic plan views of three major diffuser types: a) Unidirectional diffuser, b) Staged Diffuser, c) Alternating

diffuser. Any of those diffusers may have a variable alignment γ relative to the ambient current.

8-7
chosen and the width b is conveniently defined by the

“1/e width” where the local quantities are 1/e = 37% of

the centerline value.

When the conservation laws are applied to these four

flux quantities using a control volume of differential

length ds where s = axial direction along trajectory the

following differential equations arise:

Volume flux conservation: dQ

ds = 2π α uc b (6)

i.e. the volume flux (discharge) increases due to entrainment along the jet periphery.

Axial momentum flux conservation:

dM

ds = 2π I4 gc

′

b

2

sin θ (7)

i.e., only the sin θ component of buoyancy produces

acceleration in the axial direction, in which θ = local

vertical angle.

Horizontal momentum flux conservation:

d

ds (M cos θ) = 0 (8)

i.e., no acceleration in the horizontal direction.

Scalar flux conservation: dQc

ds = 0 (9)

Buoyancy flux conservation: dJ

ds = 0 (10)

i.e. in the uniform ambient environment both fluxes

stay constant.

In addition, it is necessary to relate the local coordinate

system ( s, θ ) to the fixed global one ( x, y )

dx

ds = cos θ (11)

dz

ds = sin θ (12)

This system of seven ordinary differential equations is

fully specified by seven initial conditions at s = 0. These

are the initial bulk fluxes Mo , Jo , Qo , and Qco (alternatively,given by Uo , go

′

 = g (ρa − ρo) ⁄ ρa , co , and D

and the geometry xo , zo , and θo.

Solution of this ordinary differential equation system by

any chosen numerical method yields the seven local

buoyant jet measures. These are M, J, Q, and Qc (or

alternatively, the related variables uc , gc

′

, cc , and b)

and the trajectory measures x, z, and θ . The local bulk

(flux-averaged) dilution is then given by the ratio

Q ⁄ Qo and the local centerline (minimum) dilution by the

ratio co ⁄ cc.

Two fundamental difficulties exist in the jet integral

method:

(i) The closure problem: Entrainment and mixing of

ambient fluid is a turbulent flow phenomenon. The

volume flux conservation, Equation 6, presupposes that

the mean entrainment velocity ve , (see Figure 8-1) is

linearly related to the centerline velocity, ve = α uc,

where α = entrainment coefficient. Inspection of data

on buoyant jets that undergo a transition from initial

jet-like (momentum-dominated) to final plume-like

(buoyancy-dominated) behavior shows that α is quite

variable. In some integral models a geometric equation

is used instead of Equation 6, namely

Jet spreading: db

ds = k (6a)

 In which k = spreading coefficient with somewhat less

variability between the jet-like and plume-like stages.

The actual choice of the appropriate equation, Eqs. 6 or

6a, and the specification of the coefficient that may be

a function of local flow conditions is generally referred

to as the “closure problem”. The closure is made differently in the various integral models. A more detailed

discussion is given by Holley and Jirka (1986).

(ii) The zone of flow establishment: The above equation

set is, strictly speaking, not valid in a short initial zone

of flow establishment in which a gradual adjustment

between the efflux profile (approximately uniform) to the

final bell-shaped profile takes place. Since this zone is

short ( ≈ 5D to 10D, where D = diameter of the discharge

port), no major error is introduced if it is simply neglected. This is the case in some integral models.

Alternately, some models include an adjustment via a

virtual origin or others perform a detailed, though approximate, analysis of this zone.

The derivation of integral jet equations for the slot

buoyant jet (see the alternative source conditions indicated in Figure 8-1) is quite analogous to the round jet.

It is omitted here for brevity (see Holley and Jirka, 1986).

The slot buoyant jet is an important element of the

analysis of subsurface multiport diffuser plumes that are

formed after merging of the individual round jets.

8.4.2 Extensions to Flowing Stratified Ambients

The advantage of jet integral models is their ready

extension to more complex environmental conditions,

such as ambient stratification and crossflow.

8-8
If the receiving water is stratified with a stable density

gradient (dρa ⁄ dz < 0, i.e. the ambient density

= ρa ( z ) decreases upward), then the buoyancy flux is

not conserved along an upward jet trajectory but is

constantly decreasing. Eventually the buoyant jet will

reach, and may even overshoot, its terminal level zt at

which the local internal jet density is equal to the

ambient density ρa ( z t ). The jet will become trapped

at this level and spread horizontally in the form of a

gravitational current. The jet mechanics prior to the

terminal level are readily described with the integral

technique if two extensions are made. First, the buoyancy profiles are now defined with respect to the local

reference buoyancy

g

′

 =

ρa (z) −ρ

ρa (z)

g (13)

instead of Equation 2, leading to modification of Equations 3 and 7, respectively. Second, from mass balance requirements, the buoyancy flux is decreasing at

the same rate at which it is diluted with ambient water

of lesser density. This leads to

d J

d s

= Q

g

ρa

d ρ a

d s

(14)

for the round jet, instead of Equation 10. Inherent in

these expressions is the assumption that the average

density of the entrained water is equal to the density at

the level of trajectory (centerline). This excludes cases

of very rapid local changes, such as steep pycnoclines

in estuaries.

When a round buoyant jet is discharged into an ambient

crossflow of velocity ua , then it will be deflected in the

direction of the crossflow. This deflection is brought

about by two force mechanisms, a pressure drag force

FD and a force Fe due to the entrainment of crossflow

momentum. Referring to Figure 8-9, this situation is

readily described in the integral analysis framework

provided that several adjustments are made. First, neglecting the horseshoe or “kidney” shape (Fischer et al.,

1979) which actually exists and assuming that the jet

may be approximated by a circular cross-section, the

velocity profile in the jet cross-section is given by the

sum of the ambient velocity component in the direction

of the trajectory, ua cos θ, and the bell-shaped jet profile. This, then, affects the definition of all jet bulk flux

variables, M, J, Q and Qc. The definition of the drag

force normal to the jet axis, and per unit length of the jet

axis, is (in kinematic units)

FD =

1

2

CD ua

2

sin2 θ (2b) (15)

in which CD is a drag coefficient (of order of unity), and

the width of the “jet body” is simply taken as 2b. The

Figure 8-9. Round buoyant jet in ambient crossflow with drag and entrainment forces (Example: vertical discharge).

8-9
entrainment force (entrainment of ambient momentum) is

Fe = ua

dQ

ds (16)

The governing momentum equations, Equations 7 and

8 are amplified to

dM

ds = 2π I4 gc

′

b

2

sin θ + Fe cos θ (17)

d

ds (M cos θ) = Fe + FD sin θ (18)

 Also, it is observed in bent-over jets that the entrainment mechanism is considerably more vigorous and

the entrainment velocity not simply proportional to uc

as in the previous case. Several analyses have suggested that jet entrainment in crossflows has a second

contribution once the jet is strongly bent-over but still

slowly rising. This second contribution is similar to that

of a horizontal line element of fluid that is rising due to

an initial vertical impulse of momentum or due to initial

buoyancy in a stagnant ambient fluid. The rising line

element experiences turbulent growth and entrainment that is proportional to the velocity of rise. Since

the strongly bent-over jet is similar to this line element,

this second entrainment mechanism can be added to

the original entrainment mechanism associated with

the excess of forward jet velocity relative to the surrounding fluid. The result is

dQ

ds = 2π α uc b + 2π α2 ua b sin θ cos θ (19)

where α is of the same form as for a buoyant jet in

stagnant ambient (Equation 6) and α2 is the crossflow

induced entrainment coefficient.

8.4.3 Overview of Jet Integral Models Available

for Mixing Zone Analysis

A large number of jet integral models for submerged

single port or multiport discharges are reported in the

literature. However, only a few of these are available

for practical mixing zone analysis in the form of computer programs accessible to the analyst. Several of

these are discussed below.

The validity and reliability of a jet integral model should

be promulgated on at least two considerations: First,

is its theoretical formulation sound and does it perform

accurately under limiting conditions (e.g. the pure jet

or pure plume)? Second, how do the model predictions compare with available data, preferably field

data? No complete evaluation on these grounds of

integral jet models is attempted here, but some important model features will be addressed in Section 3. It is

stressed again that none of the following integral jet

models include any form of boundary interaction processes; in a sense they all assume an unlimited receiving

water body.

The U.S. EPA has published a set of five buoyant jet

integral models (Muellenhoff et al., 1985), all with different capabilities. These models include computer programs written in FORTRAN for micro or minicomputers.

(1) The computer model UPLUME describes a buoyant

jet issuing from a single port into a stagnant environment

with arbitrary stratification. UPLUME is based on Abraham’s (1963) original development using a jet spreading

equation for closure. Empirical adjustment expressions

are included for the zone of flow establishment.

(2) The model UOUTPLM (based on Winiarski and

Frick, 1976) uses a somewhat different Lagrangian

description of buoyant jet mechanics instead of the

Eulerian system of equations given in Section 8.4.1.

Thus, a plume element is tracked in its time-dependent

evolution. However, the mechanisms actually included

are similar to the ones discussed above with the exception of the omission of the ambient drag force. The

model is applicable to a uniform crossflow with co-flowing or cross-flowing single port orientation (excluding

counterflows) and with arbitrary density stratification.

The model is not applicable for stagnant conditions.

(3) The model UMERGE is an extension of UOUTPLM

applicable to multiport diffusers with perpendicular

alignment. Merging is assumed to occur when geometric overlap of the individual equally spaced round jets

occurs. After merging, the flow is described by the

time-dependent motion of two-dimensional plume elements.

(4) UDKHDEN is a model that computes three-dimensional trajectories from either single port or multiport

discharges in crossflows with arbitrary velocity (shear

flow) and density distributions. The model is based on

the development by Hirst (1971) and later generalizations by Kannberg and Davis (1976). The initial zone of

flow establishment is computed in detail with Hirst’s

model. The three-dimensional equation system is a

generalization of the type discussed in the preceding

section. An entrainment function with dependence on a

local densimetric Froude number is used for closure. A

special geometric merging routine describes the gradual transition from individual round plumes to the twodimensional plume. However, the same entrainment

coefficient is used for round and for plane buoyant jets,

making it impossible to verify the model

8-10
for well-known asymptotic conditions. The diffuser

alignment relative to the crossflow must be predominantly perpendicular.

(5) The model ULINE is strictly speaking not a jet

integral model but uses an analytical solution for the

two-dimensional slot plume dilution as a function of

elevation. This solution is modified on the basis of

Roberts’ (1977) experimental results for the effect of

alignment on a diffuser line plume in crossflow. Also a

stepwise algorithm is included to compute local mixing

in an arbitrary crossflow and stratification. The model

omits the merging process, thus assuming an initially

merged (e.g. closely spaced) diffuser discharge.

Another buoyant jet model is that of Jirka and Fong

(1981) to predict general three-dimensional trajectories for a single port discharge in a crossflow with

arbitrary stratification. The model uses empirical descriptions for the zone of flow establishment as proposed by Schatzmann (1978). The model includes an

entrainment closure that meets several limiting conditions and that has been extensively verified by Wong

(1984) in application to ambient stratification. An additional element of the Jirka-Fong model is the description of the internal vortex mechanism in crossflow that

can lead to plume bifurcation when a flow boundary or

terminal level is encountered.

8.5 CORMIX: Expert System Methodology

for Mixing Zone Analysis

8.5.1 Introduction

The Cornell Mixing Zone Expert System (CORMIX) is

a series of software elements for the analysis and

design of submerged buoyant or nonbuoyant discharges containing conventional or toxic pollutants into

stratified or unstratified watercourses, with emphasis

on the geometry and dilution characteristics of the

initial mixing zone. Subsystem CORMIX1 (Doneker

and Jirka, 1990) deals with single port discharges and

subsystem CORMIX2 (Akar and Jirka, 1991) addresses multiport diffusers [Another subsystem, CORMIX3 (Jones and Jirka, 1991), has been developed for

surface discharges, but is not discussed here given the

limitations of surface discharges in meeting toxic dilution criteria; see Chapter 7]. The system is implemented on microcomputers with the MS-DOS

operating system.

The user supplies CORMIX with information about the

discharge and ambient environment. CORMIX returns

information detailing the hydrodynamic mechanisms

controlling the flow, dilution, geometric information

concerning the shape of the pollutant plume or flow in

the ambient water body, and design recommendations

allowing the user to improve the dilution characteristics

of the flow. If specified by the user, CORMIX also

presents information about legal mixing zone dimensions and dilution and about toxic mixing zone requirements.

CORMIX contains two key elements. The first is a

rigorous flow classification scheme that classifies any

given discharge/environment situation into one of several flow classes with distinct hydrodynamic features.

The classification scheme places major emphasis on

the near-field behavior of the discharge and uses the

length scale concept as a measure of the influence of

each potential mixing process. Flow behavior in the

far-field, mostly in the form of boundary interactions, is

also considered.

The second key element is a collection of predictive

elements (modules) that are executed according to a

protocol that pertains to each distinct flow class as

determined by the classification scheme. These predictive elements are all based on simple analytical perturbation solutions for each flow process. Furthermore,

transition rules are used to describe the spatial extent

of each flow process.

The final result is a robust composite flow and mixing

zone prediction that is applicable to a diverse variety of

discharge/ambient conditions. CORMIX1 and 2 have

been extensively validated with both laboratory and field

data.

The geometric schematizations assumed in CORMIX

have been summarized in Figures 8-6 to 8-8, respectively. In addition, CORMIX assumes a uniform unsheared ambient velocity profile represented by the

mean velocity ua. Furthermore, CORMIX requires that

the ambient density profile be approximated by one of

four representative stable profiles as shown in Figure

8-10. A dynamically correct approximation of the actual

distribution should keep a balance between over- and

under-estimation of the actual density data. The simplest case is a linear density profile shown in Figure

8-10a (Stratification Type A). Figure 8-10b describes

two uniform density layers with a density jump (pycnocline) between layers (Stratification Type B). Figure

8-10c illustrates a two layer profile in which the upper

layer is uniform, the lower layer has a linear stratification, and a density jump occurs between layers (Stratification Type C). Finally, Figure 8-10d presents a two

layer system with a uniform upper layer and a linearly

stratified bottom layer with no density jump between

layers (Stratification Type D). The uniform upper layers

in Stratification Types B, C, or D are representative for

the well mixed upper layer that is found in many types

of ambient water bodies and occurs due to wind induced

turbulent mixing.

8-11
8.5.2 Length Scales

Length scales, obtained from dimensional analysis,

describe the relative importance of discharge volume

flux, momentum flux, buoyancy flux, ambient crossflow, and density stratification in controlling flow behavior. The length scales will describe the distance over

which these dynamic quantities control the flow, in

particular within the subsurface buoyant jet regions of

the mixing process.

8.5.2.1 Single Port Discharges

Given the important flux parameters, Qo , Mo , and

Jo (see Figure 8-5), the ambient velocity ua, and the

buoyancy gradient ε = − ( g ⁄ ρa ) ( d ρa ⁄ d z ) of a linearly stratified ambient, the following dynamic length

scales can be derived for a single port discharge:

LQ = Qo ⁄ Mo

1

⁄2= discharge (geometric) scale

LM = Mo

3

⁄4

⁄ Jo

1

⁄2 = jet/plume transition scale

Lm = Mo

1

⁄2

⁄ ua = jet/crossflow scale

Lb = Jo ⁄ ua

3

 = plume/crossflow scale

Lm

′

 = (Mo ⁄ ε )

1

⁄4

 = jet/stratification scale

Lb

′

 = Jo

1

⁄4

⁄ ε

3

⁄8

 = plume/stratification scale

The meaning of these scales is further illustrated in

Figure 8-11. For example, the jet/crossflow length

scale is a measure for the distance over which a pure

jet will intrude into a crossflow before it gets strongly

deflected (or affected). It should be noted that the

length measures are only “order of magnitude”; precise coefficients have to be determined from experiments or from more detailed flow analysis.

8.5.2.2 Multiport Diffusers

The general diffuser flow field is, of course, three-dimensional. However, for near-field mixing analyses the twodimensional flow parameters are dynamically relevant.

For this purpose, the details of individual discharge jets

with port diameter D and spacing S are neglected and

replaced by an equivalent slot width B =(πD

2

) ⁄ (4 S ) on

the basis of equivalency of momentum flux per unit

diffuser length. This concept has been discussed by

Jirka (1982b) among others, and has been shown to be

a dynamically accurate representation. The main parameters for the two-dimensional slot discharge are the

diffuser total flowrate Qc and the discharge buoyancy

go

′

 . This leads to the following flux parameters (per unit

diffuser length), all expressed in kinematic units:

qo = Qo ⁄ LD = volume flux (flowrate)

mo = qo Uo = Uo

2 B = momentum flux

jo = qgo

′

 = Uogo

′ B = buoyancy flux,

 in whichUo = discharge velocity, and

LD = diffuser length.

Through interaction with the ambient parameters, the

following length scales describe a multiport diffuser

discharge:

lq = q o

2

⁄ mo = discharge geometric scale

lm = mo ⁄ uα

2

 = plane jet/crossflow scale

Figure 8-10. Schematic ambient density profiles for use in expert system CORMIX.

8-12
Figure 8-11. Length scales measuring the effects of momentum flux, buoyancy flux, crossflow and stratification of submerged 

8-13
lM = mo ⁄ jo

2⁄3 = plane jet/plane plume scale

lm

′

 = (mo ⁄ ε)

1

⁄3

 = plane jet/stratification scale

lb

′

 = jo

1

⁄3

⁄ ε

1

⁄2

 = plane plume/stratification scale

la = ua ⁄ ε

1

⁄2

 = crossflow/stratification scale

It is interesting to note that no plume/crossflow length

scale can be defined on dimensional grounds for the

two-dimensional plume. This is in contrast to the threedimensional round plume and arises from the fact that

the vertical velocity of a two-dimensional plume is

constant, ~j o

1

⁄3

, leading in the presence of a constant

crossflow to a straight-line trajectory. Thus, no distinction can be made of a plane plume in a weakly deflected stage followed by a strongly deflected stage.

However, it is possible to define a non-dimensional

parameter j o ⁄ U a

3 whose magnitude will be a measure

of the slope of the plume trajectory.

8.5.3 Near-Field Flow Classification

The classification scheme used in CORMIX puts major

emphasis on the near-field flow configuration. This is

because a large number of flow configurations can

occur due to the multiplicity of possible interaction

processes; in contrast the far-field flow is generally

much simpler with limited shoreline or bottom contact

possibilities.

8.5.3.1 Single Port Discharges (CORMIX1)

In the near-field the dynamic length scales

LM , Lm , Lb , Lm

′

and Lb

′

 (LQ has less significance) describe the interaction with the geometric properties of

the water body, its depth H or the depth hint to the

density jump (in general, both of those are indicated

by a layer depth Hs ). Also the orientation angles θo

and σo of the discharge are important (Figure 8-6).

Given the possible ambient stratification types a classification procedure (in Doneker and Jirka, 1990) is

used to classify the near-field behavior of a given

discharge into one of 35 generic flow classes that are

summarized in Figures 8-12 to 8-15. The four major

flow categories indicated by CORMIX1 are:

i) flows affected by linear stratification leading to internal trapping (S classes, Figure 8-12)

ii) buoyant flows in a uniform ambient layer (V and H

classes, Figure 8-13)

iii) negatively buoyant flows in a uniform ambient layer

(NV and NH classes, Figure 8-14)

iv) bottom attached flows (A classes, Figure 8-15).

Each of the flow classes is indicated on the figures by

a sketch that shows its main features in a side view or

plan view. All flow criteria shown on the figures are given

as “order of magnitude” relations; somewhat different

forms and numerical constants may be contained in

CORMIX1.

A wide spectrum of near-field flow configurations is

possible: these range from flows trapped in linear stratification, buoyant jets that are strongly affected by the

crossflow and gradually approach the layer boundary

(surface or pycnocline), weakly deflected buoyant jets

that impinge on the boundary leading to upstream

spreading and/or unstable recirculation, negatively

buoyant jets that form density currents along the bottom, and dynamic attachment along the bottom with or

without eventual buoyant lift-off. It is stressed also that

(i) each of these flow classes can occur in combination

with an upper stratified layer (see stratification types B,

C, or D on Figure 8-10) and (ii) the designation “uniform

ambient layer” in Figures 2-13 and 2-14 can, in fact, also

apply to a stratified layer if it has been found that the

stratification is too weak to trap the flow. Thus, in

essence, the actual number of flow configurations that

can be classified by CORMIX1 is much larger than the

35 generic flow classes shown on these figures.

8.5.3.2 Multiport Diffusers (CORMIX2)

The classification scheme used by CORMIX2 relies on

the same methodology as for single port discharges.

The length scales of the two-dimensional slot jet,

IM , Im , Ib , lm

′

, and Ia , are compared with the layer

depth Hs

 and with the diffuser variables, its length LD

and its orientation angles, θ, γ, β, σ (see Figure 8-7).

The classification procedure (see Akar and Jirka, 1991,

for details) yields 31 generic flow classes that fall into

three major categories: (i) flows affected by linear stratification leading to internal trapping (MS classes, Figure

8-16), ii) buoyant flows in uniform ambient layers (MU

classes, Figure 8-17), and iii) negatively buoyant flows

in uniform ambient layers (MNU classes, Figure 8-18).

While there are some obvious analogies in their appearance to the flows produced by single port discharges,

the major difference for multiport diffusers lies in the

vertically fully mixed (over the layer depth) plumes that

can be produced by the large momentum sources of

unidirectional or staged diffusers.

8.5.4 Predictive Elements

The detailed hydrodynamic prediction of the effluent

flow and of associated mixing zones in CORMIX is

carried out by appropriate flow modules that are ex8-14
Figure 8-11. Length scales measuring the effects of momentum flux, buoyancy flux, crossflow and stratification of submerged 

8-15
Figure 8-13. CORMIX1 sub-classification: flow classes for positively buoyant single port discharges in uniform ambient 

8-16
Figure 8-14. CORMIX1 sub-classification: flow classes for negatively buoyant single port discharges in uniform ambien

8-17
Figure 8-15. CORMIX1 sub-classification: assessment of dynamic bottom attachment processes and flow classes for bottom-a

8-18
Figure 8-16. CORMIX2 sub-classification: assessment of density stratification and flow classes for internally trapped multipor

8-19
Figure 8-17. CORMIX2 sub-classification: flow classes for positively buoyant multiport discharges in uniform ambient 

8-20
Figure 8-18. CORMIX2 sub-classification: flow classes for negatively buoyant multiport discharges in uniform ambient 

8-21
ecuted according to a protocol that pertains to each

distinct flow configuration as determined by the classification scheme. These flow protocols have been constructed on the basis of the same length scale

arguments that have been used for the flow classification. The spatial extent of each flow module is governed by transition rules. These determine transitions

between different near-field and far-field mixing regions, and distances to boundary interaction.

The flow modules for single port discharge predictions

(CORMIX1) are listed in Table 8-1. All modules present basic analytical solutions for one particular flow

process with the perturbing influence of one or more

other variables superimposed. For example, the module for the weakly deflected jet in crossflow (MOD11)

is based on a pure jet solution that experiences a

gradual advection by the crossflow. The group of nearfield modules (MOD01 to MOD22) represents, in total,

the same predictive ability as buoyant jet integral models (valid in the subsurface region without boundary

interaction).

The flow modules for multiport diffuser prediction

(CORMIX2) are given in Table 8-2. Several groups of

modules, notably those for the far-field, are similar, or

even identical, to those of CORMIX1.

Extensive comparisons have been conducted for CORMIX1 and 2 with available laboratory data and a few

limited field data cases, as well as with buoyant jet

integral models. These comparisons (Doneker and

Jirka, 1990; Akar and Jirka, 1991) demonstrate that for

subsurface flow the CORMIX predictions were at least

of the same quality as that of jet integral models. The

agreement with data (±20% for trajectories and dilutions) is of the same order as the usual scatter among

different data sources.

Modules for Buoyant Jet Near-Field Flows

zone of flow establishment

weakly deflected jet in crossflow

weakly deflected wall jet in crossflow

near-vertical jet in linear stratification

near-horizontal jet in linear stratification

strongly deflected jet in crossflow

strongly deflected wall jet in crossflow

weakly deflected plume in crossflow

strongly deflected plume in crossflow

Modules for Boundary Interaction Processes

near-horizontal surface/bottom/pycnocline approach

near-vertical surface/bottom/pycnocline impingement

 with buoyant upstream spreading

near-vertical surface/bottom/pycnocline impingement with

 vertical mixing

near-vertical surface/bottom/pycnocline impingement,

 upstream spreading, vertical mixing, and buoyant

 restratification

terminal layer stratified impingement/upstream spreading

terminal layer injection/upstream spreading

Modules for Buoyant Spreading Processes

buoyant layer spreading in uniform ambient

buoyant spreading in linearly stratified ambient

Modules for Attachment/Detachment Processes

wake recirculation

lift-off/fall-down

Modules for Ambient Diffusion Processes

passive diffusion in uniform ambient

passive diffusion in linearly stratified ambient

Table 8-1. Flow Prediction Modules of CORMIX1 (Single Port

Discharges)

Simulation Modules for Buoyant Multiport Diffusers:

Subsurface Near-Field Flows

discharge module

discharge (staged diffuser)

weakly deflected plane jet in crossflow

weakly deflected (3-D) wall jet in crossflow

near-vertical plane jet in linear stratification

near-horizontal plane jet in linear stratification

strongly deflected plane jet in crossflow

weakly deflected (2-D) wall jet in crossflow

weakly and strongly deflected plane plume in crossflow

buoyant plane plume in stratified stagnant ambient

negatively buoyant line plume

Simulation Modules for Unstable Multiport Diffusers:

Mixed Near-Field Flows

unidirectional acceleration zone

tee acceleration zone

strongly deflected tee diffuser plume

staged acceleration zone

strongly deflected staged diffuser plume

alternating perpendicular diffuser in unstable

 near-field zone

negatively buoyant staged acceleration zone

Simulation Modules for Boundary Interaction Processes

 for Stable Multiport Diffusers

near-vertical surface/bottom impingement with buoyant

 upstream spreading

near-vertical surface/bottom impingement, upstream

 spreading, vertical mixing, and buoyant

 restratification

near-horizontal surface/bottom/pycnocline approach

terminal layer stratified impingement/upstream spreading

terminal layer injection/upstream spreading

Simulation Modules for Unstable Multiport Diffusers:

Intermediate Field Flows

diffuser plume in co-flow

diffuser plume in crossflow

Simulation Modules for Buoyant Spreading Processes

buoyant layer spreading in uniform ambient

buoyant spreading in linearly stratified ambient

density current developing along diffuser line

internal density current developing along diffuser line

diffuser induced bottom density current (2-D)

diffuser induced bottom density current (3-D)

Simulation Modules for Ambient Diffusion Processes

passive diffusion in uniform ambient

passive diffusion in linearly stratified ambient

Table 8-2. Flow Prediction Modules of CORMIX2 (Multiport

Diffusers)

8-22
Moreover, CORMIX has been shown to be a robust

and accurate predictive methodology for more complex flows with various degrees of boundary interaction, such as near-field instabilities, buoyant spreading

processes, and dynamic bottom interaction. CORMIX

appears to correctly diagnose these processes

through its classification scheme and then provides

quantitatively reliable predictions of the sequence of

mixing processes that characterize a given discharge.

However, as all models that are based on some geometric schematizations and dynamic simplifications,

CORMIX will not be applicable to all possible discharge

configurations. To avoid model misuse in such instances, specific safeguards, warning labels, and use

restrictions have been included in CORMIX. In any

case, recent experience has shown that CORMIX is

applicable and predicts properly for the vast majority

of actual submerged discharge situations (better than

95% for CORMIX1 and better than 80% for CORMIX2

because of the considerably greater geometric complexities of diffuser installations). Furthermore, the

user’s manuals contain special advice sections for the

user dealing with any of the more limiting cases.

8.6 Mixing Zone Predictions Under

Unsteady Reversing Tidal Currents

As has been remarked earlier in Section 8.1, the time

scale for initial mixing processes is usually short

enough relative to the tidal period, so that it is acceptable to apply initial mixing models under steady-state

conditions, e.g. corresponding to certain stages within

the tidal cycle. However, this approach is no longer

valid if predictions are desired over a larger area

encompassing distances that, in fact, provide a transition to the far-field.

In the present state-of-the-art no complete models for

pollutant predictions in the water environment are

available (see Section 8.2). This restriction stems from

the difficulties of representing the variety of transport

processes that govern the distribution in unconfined

estuarine or coastal water bodies in a single analytical

or numerical technique. Therefore, an integration of

near-field mixing models and of predictive techniques

for the far-field effects must be employed. Far-field

processes, that include the transport by the varying

tidal flow, turbulent diffusion, and various biochemical

transformation phenomena, have been addressed in

Parts I and II of this estuarine waste load allocation

manual. The following comments provide some guidance on estimating, the interaction between near-field

mixing and far-field accumulation effects. The methodology is adapted from that suggested by Jirka et al.

(1976).

8.6.1 Far-Field Accumulation Effects

The two major methods for estimating the unsteady

far-field accumulation of discharged material, at variable distances from the outfall and in an unsteady tidal

flow, are either numerical models or field dispersion

tests. In the following it is assumed that a dispersion

test is being employed, but the comments apply

equally well to the results of an unsteady numerical

model.

The schematics of a field dispersion test in a reversing

tidal current system are shown in Figure 8-19. The

tracer release line may represent the location of a

submerged multiport diffuser with alternating nozzles.

The tidal system is assumed as approximately periodic

as indicated by the velocity curve. The figure also

shows the hypothetical dye concentration trace C(x,y)

measured at some point ( x , y ) as a function of time.

(Note that in practice, fewer discrete measurements

over time would be available). If the field dispersion

test consists of a tracer release period, tidal cycles

long, then the continuous monitoring would usually

indicate a period of concentration build-up, a quasisteady period and a fall-off period. If an accurate

simulation of the pollutant discharge over a large-scale

and for a long-term is required, then consideration (and

measurement) for at least two of these periods is

necessary.

Considering the maximum dye concentration during

any tidal cycle at ( x , y ) the following sequence is

generally observable: During the first cycle C max is

found, in the second cycle the concentration is

plus some fraction of dye tracer returning from the

previous cycle, thus C max + rd C max = C max (1+rd ).

If these are continuously repeated, then the quasisteady maximum concentration C

__

max is given by the

geometric series

C

__

max = C max ( 1 +r d + rd

2 + rd 3 + 〈) (20)

or, in the limit,

C

__

max = C max 1

1−rd

(21)

The quantity rd is labelled the dye return rate of mass

discharged in the previous cycle (rd implicitly includes

any dye mass decay during the tidal period). The

complement quantity (1-rd ) is frequently referred to

as flushing rate. The return rate will depend on the

characteristics of the tidal flow, notably tidal excursion,

mean velocity, diffusion, etc. rd is also dependent on

the position ( x , y ) with respect to the release area.

Quasi-steady conditions are typically encountered after about 5 to 10 tidal cycles. Build-up curves, similar

to Equation 20 correspond also to other quantities of
interest, such as the minimum or average concentrations during a tidal cycle, thus

C

__

i ( x , y , t ) = Ci ( x , y , t ) 1

1−rd

 (22)

where Ci ( x, y )is a single cycle concentration quantity

of interest (Cmax ,Cmin ,Cavg , etc.).

For the actual pollutant discharge the quasi-steady

condition is usually of primary importance. From Equation 22 it is seen that this depends on two factors: the

mixing characteristics Ci within a single tidal cycle, and

the return rate from previous cycles. To translate the

quasi-steady dye concentration conditions into pollutant concentration, therefore, two adjustments are

needed:

(a) Within a tidal cycle, the pollutant concentration c is

related to the dye concentration C

ci ( x, y, t ) = Ci ( x, y, t ) Qco

Qdo

e

− (kc−kd) ti ( x, y ) (23)

where ti ( x, y ) = time interval between occurence of

event i (maximum, minimum concentration) at

( x , y ) and time of release of that tracer patch, i.e.,

travel time. Qco is the pollutant mass release rate and

Qdo is the dye mass release rate. kc and kd represent

the decay constants for pollutant and dye, respectively. (for a conservative dye, kd=0). Determination of

ti depends on the detailed knowledge of the velocity

field; for average concentrations the average tidal

velocity is representative. It is noted that for points far

from the release area, especially more than several

tidal excursions away, the exponential correction term

in Equation 23 becomes significant. In the discharge

vicinity, however, it is frequently negligible, since ti is

less than one tidal period. This is, in fact, the usual

assumption in most mixing zone predictions.

(b) The return rate for pollutant rc is related to the dye

return rate rd

rc = rd e

−(kc−kd) t

∗

(24)

where t

∗

= tidal period (12.4 hours). The quasi-steady

pollutant concentration c

_

i ( x,y ) is therefore related to

the measured single cycle dye concentration

Ci ( x,y )

c

_

i ( x , y , t ) = Ci ( x , y , t )Qco

Qdo

 [ e

− (kc−kd) ti

]

1−rd

1−rc

(25)

Hence, for an accurate prediction of far-field effects

over a large area (larger than the tidal excursion

length) it is necessary to (i) measure the velocity field

in some detail so ti ( x, y ) can be found for the points

under consideration, and (ii) measure not only the

quasi-steady period of tracer distribution, but also the

build-up or fall-off period so the dye return rate rd can

be evaluated as shown in Figure 8-19. In actual tracer

monitoring it is not always possible to have continuous

records. Nevertheless, a few measurements during

the build-up or fall-off period usually give some indication of rd.

If attention is restricted to a smaller area around the

discharge and if the tracer used is relatively conservative (small kd), then both correction factors in Equation

25 are negligible and the measured concentrations

can be used directly to evaluate the pollutant accumulation in the far-field.

8.6.2 Linkage to Initial Mixing Predictions

All initial mixing models discussed in the preceding are

steady-state models and do not consider the far-field

return (accumulation). The following procedure provides an approximate linkage:

(a) Carry out a series of initial mixing predictions using

a steady-state near-field mixing model for different

intervals (e.g. 6 or 12, corresponding to 2 or 1-hour

intervals, respectively) within the tidal cycle. The predictions at any point of interest (e.g. at the boundary of

a Legal Mixing Zone) provide approximate time-dependent predictions for pollutant concentration

ci ( x, y, t ) within a tidal cycle.

(b) Use the far-field pollutant return rate rc, that applies

for the region of interest (e.g. the Legal Mixing Zone),

to calculate the quasi-steady (i.e. long-term) pollutant

concentration

c

_

i ( x , y , t ) = ci ( x , y , t ) 1

1−rc

 (26)

The return rate rc that applies to the area of interest

can be estimated using the procedures outlined in the

preceding paragraph, i.e. relying on a dye dispersion

test or numerical model. It should be noted that rc, in

turn, is a function of the distance from the outfall: rc

tends to be very small in the immediate near-field,

where the pollutant concentrations are high; rc becomes larger for increasing distances, where the induced concentrations are falling off, however. This

dependence suggests the following practical guidelines in the absence of detailed measurements or

predictions for rc :

8-24
Figure 8-19. Schematics of a field tracer dispersion test in a periodically reversing tidal system.

8-25
• For Toxic Dilution Zone (TDZ) predictions, the

effect of far-field return is always negligible (rc ≈ 0)

due to the strong spatial restriction of the TDZ.

• For most Legal Mixing Zone predictions, the rc

factor can be expected to vary in the range of ≤0.1

to ≈ 0.5 (highly conservative estimate). It is very

small (≤ 0.1) for deep water discharges in the open

coastal zone that are often associated with internal

trapping or buoyant surface layer formation. In

those cases, the initial (buoyant jet) mixing is, in

fact, quite independent of far-field effects. It may

be reasonably high (up to 0.5) for shallow water,

vertically mixed, discharges in strongly restricted

estuaries with weak flushing. For additional flushing estimates in such tidal channels, see the methods discussed in Fischer et al. (1979).

8.7 References

Abraham, G. 1963. Jet Diffusion in Stagnant Ambient

Fluid. Publ. No. 29, Delft Hydraulics Laboratory, The

Netherlands.

Akar, P. J., and G. H. Jirka. 1991. CORMIX2: An

Expert System for Hydrodynamic Mixing Zone Analysis of Conventional and Toxic Submerged Multiport

Diffuser Discharges. Technical Report, U.S. EPA, Environmental Research Laboratory, Athens, GA, (in

preparation).

Doneker, R. L., and G. H. Jirka. 1990. CORMIX1: An

Expert System for Hydrodynamic Mixing Zone Analysis of Conventional and Toxic Submerged Single Port

Discharges. Technical Report EPA 600/3-90/012, U.S.

EPA, Environmental Research Laboratory, Athens,

GA.

Fischer, H. B. et al. 1979. Mixing in Inland and Coastal

Waters. Academic Press, NY. 483 pp.

Hirst, E.A. 1971. Analysis of Buoyant Jets Discharged

to Flowing Stratified Ambients. Rep. ORNL-TM-4685,

U.S. Atomic Energy Commission, Oak Ridge National

Lab., Oak Ridge, TN.

Holley, E. R. and G. H. Jirka. 1986. Mixing in Rivers,

Technical Report E-86-11, U.S. Army Corps of Engineers, Washington, DC.

Jirka, G. H. 1982a. Turbulent Buoyant Jets in Shallow

Fluid Layers. in Turbulent Jets and Plumes, W. Rodi

(Ed.), Pergamon Press.

Jirka, G. H. 1982b. Multiport Diffusers for Heat Disposal -A Summary. Journal of the Hydraulics Div.,

ASCE, Vol. 108, December.

Jirka, G. H., G. Abraham, and D.R.F. Harleman. 1976.

An Assessment of Techniques for Hydrothermal Prediction. Technical Report NUREG-0044, U.S. Nuclear

Regulatory Commission, Washington, DC.

Jirka, G. H. and L.M. Fong. 1981. Vortex Dynamics

and Bifurcation of Buoyant Jets in Crossflow. Journal

of the Engineering Mechanics Division, American Society of Civil Engineers, Vol. 107, No. EM3, June.

Jones, G.R. and G.H. Jirka. 1991. CORMIX3: An

Expert System for the Analysis and Prediction of Buoyant Surface Discharges, Technical Report, DeFrees

Hydraulics Laboratory, School of Civil and Environmental Engineering, Cornell University. Also to be

published by U.S. Environmental Protection Agency,

Technical Report, Environmental Reasearch Lab, Athens, GA.

Kannberg, L. D., and L.R. Davis. 1976. An Experimental/Analytical Investigation of Deep Submerged Multiple Buoyant Jets. EPA-600/3-76-101. U.S.

Environmental Protection Agency, Corvallis, OR. 266

pp.

Muellenhoff, W. P., et al. 1985. Initial Mixing Characteristics of Municipal Ocean Discharges (Vol. 1&2).

U.S.EPA. Environmental Research Laboratory, Narragansett, R.I.

Roberts, P.J.W. 1977. Dispersion of Buoyant Waste

Discharge from Outfall Diffusers of Finite Length. Rep.

No. KH-R-35. W. M. Keck Lab. of Hydraulics and

Water Resources, California Institute of Technology,

Pasadena, CA, 183 pp.

Schatzmann, M. 1978. The Integral Equations for

Round Buoyant Jets in Stratified Flows. J. Appl. Math

and Physics 29: 608-20.

USEPA. 1982. Revised Section 301 (h) Technical

Support Document. EPA 430/9-82-011, Washington,

DC.

Winiarski, L. D., and W.E. Frick. 1976. Cooling tower

plume model. EPA-600/3-76-100. U.S. Environmental Protection Agency, Corvallis, OR.

Wong, D. R. 1984. Buoyant Jet Entrainment in Stratified Fluids. Ph.D. Thesis, Civil Engineering Dept., The

University of Michigan, Ann Arbor, MI.

8-26
9. Case Studies of Mixing Zone Prediction

9.1 Introduction

9.1.1 Objectives

This case study section has several objectives: (i) To

demonstrate the typical procedures and data requirements involved in mixing zone analysis; (ii) To demonstrate that legal mixing zone definitions may require

the analysis of both near-field and far-field processes;

and (iii) To show the relative merits and flexibility of

different methodologies, including jet integral models

and the expert system CORMIX.

All four case studies deal with hypothetical conditions

that may, however, exhibit some features of existing

discharges. In the first case study major emphasis is

put on various regulatory criteria. None of the case

studies is intended to document model validation. This

cannot be done since no actual field or laboratory data

exist for these hypothetical situations. For validation of

models reference should be made to the original literature on the various models as listed in Chapter 8.

However, a few comments on model validity are made

in the first case study in order to explain some large

differences in various model predictions.

9.1.2 Data Needs

As discussed in Section 8.1, the initial mixing of an

effluent depends on the interaction of ambient and

discharge conditions. In estuaries or coastal waters

these conditions may be highly variable. In evaluating

water quality effects and mixing zone compliance,

appropriate design conditions must be chosen. Generally, the critical design conditions relate to those

environmental and discharge factors that lead to the

lowest dilution and at times when the environment is

most sensitive. However, it is not always straightforward for the analyst to estimate exactly what combination of factors will lead to this critical condition. For this

reason, an evaluation under a variety of conditions

always seems necessary to obtain information on mixing zone behavior and its sensitivity to design criteria.

Data uncertainty is also a factor of concern. The following considerations, taken from Muellenhoff et al.

(1985), apply here:

“Predicting dilution reliably depends on the availability

of statistically valid data with which to estimate ambient

conditions. The statistical uncertainty in estimates of

absolute worst case conditions is generally great. Also

there are inherent biases to some oceanographic

measurements. For example, current measuring instruments have finite thresholds. It therefore becomes

difficult to distinguish low values (which may be as high

as 5.0 cm/sec) from zeroes in these data sets. In estimating environmental conditions, a more reliable estimation can be made at the lowest 10 percentile on a

cumulative frequency distribution. Data on ambient density structure are not routinely collected. Consequently,

there is not usually an existing data set for the site under

consideration. To increase the reliability of ‘worst-case’

estimates, data should be evaluated not only for the

discharge site but for nearby coastal areas of similar

environmental setting.”

“Defining ‘worst-case’ conditions as a combination of

those conditions affecting initial dilution, each taken at

the worst 10 percentile on cumulative frequency distributions, is recommended by USEPA. This approach

allows a reliable estimation of these conditions to be

made and prevents the unlikely occurence of more

extreme conditions from biasing the predictions. The

probability of these conditions occurring simultaneously

is much less than 10 percent, ensuring that the predicted

dilution will be exceeded most of the time. Application of

multiple ‘worst case’ factors (i.e. flows, stratification and

currents) to determine a minimum dilution must be done

carefully, however, and in recognition of the criteria for

which compliance is being determined. For example,

although application of an absolute ‘worst case’ dilution

may be appropriate for determining compliance with an

acute toxicity limit, it is more appropriate to identify the

lowest 6-month median dilution to determine compliance with a 6-month median receiving water limitation.”

Since the discharge conditions can also vary (e.g. its

flowrate or pollutant concentration) it is necessary to

combine the occurences of the varying pollutant loading

with the varying ambient parameters in order to find the

critical design conditions.

Finally, any set of ambient and discharge conditions will

require some degree of schematization in order to meet

the predictive model assumptions. This has been discussed in Section 8.3.2 along with Figures 8-6, 8-7 and

8-8. The literature or user’s manuals for the various

models usually contain some guidance on how to prepare the data. As with any model application, it is necessary to evaluate the prediction sensitivity to input data

through repeated model use. The expert system CORMIX, in fact, has on-screen advice on data preparation

available to the user.

All available mixing zone models assume a conservative

pollutant discharge neglecting any physical, chemical or

biological decay or transformation processes. For most

substances this is reasonable due to the
rapidity of the mixing process, especially in the nearfield, relative to the reaction time scale of most pollutants. If first order reaction processes can be assumed

then the model results on concentration can usually be

converted with an exponential factor to include the

decay process (see Doneker and Jirka, 1990). The

consideration of pollutant reactions in the context of

far-field accumulation involving a larger time scale has

also been addressed in Section 8.6.1.

9.2 Case AA -Single Port Discharge:

Industrial Outfall in Tidal Fjord.

9.2.1 Ambient and Discharge Conditions

A manufacturing plant is located near the upstream

end of a narrow tidal fjord that receives a substantial

amount of fresh water inflow. The typical cross-section

of the fjord is 600 m wide with an average depth of

16 m. The preferred discharge location is about 90 m

from shore where the local water depth is 17.5 m.

During typical winter conditions the characteristic ambient (average tidal) velocity is 0.15 m/s and the vertical ambient density distribution is quite uniform with a

value of 1,005.5 kg/m3

. During summer design conditions, however, the ambient velocity is lower at 0.10

m/s and a significant vertical stratification exists as

shown in Figure 9-1. The density varies from a bottom

value of 1,010.0 kg/m3 down to a surface value of

1,005.8 kg/m3

. The plant operation is also variable. In

winter the discharge flow rate is 0.15 m3

/s and has a

discharge temperature of 10°C. In summer the flow

rate is lower at 0.10m3

/s with a temperature of 15°C.

The discharge flow is essentially freshwater but contains 1000 ppb of some organic toxic material.

Applicable state regulations limit the mixing zone to 25%

of the width of the estuary. Furthermore, the special

mixing zone requirements for toxic substances (see

Section 7.2.3) apply with a CMC value of 100 ppb for

the discharged toxicant.

9.2.2 Case AA1: Initial Design, Winter Conditions

An inital design proposal calls for a single port discharge

with 0.2m port diameter and 0.5 m port height above the

bottom. The discharge velocity is 4.8 m/s. The port is

oriented in a co-flowing arrangement pointing horizontally along the direction of the ambient current.

Figure 9-2 shows a side view of the near-field of the

discharge plume predicted by CORMIX1 (flow class A5).

The model shows strong dynamic attachment of the

plume to the bottom. After this a gradual buoyant rise to

the surface takes place with a minimum surface dilution

= 164. The extent of the toxic dilution zone (TDZ)

is about 10m, essentially comprising the entire bottom

attached zone. Thus, benthic organisms will be exposed

to toxicant concentrations above CMC values. This initial design is considered undesirable and rejected from

further consideration.

In view of this bottom attachment, none of the jet integral

models, included in Section 8.4, i.e. the USEPA models,

UOUTPLM and UDKHDEN or the Jirka-Fong model,

would be applicable. Therefore, their predictions are not

shown on Figure 9-2.

9.2.3 Case AA2: Modified Design, Winter

Conditions

In order to eliminate plume bottom interference, a modified design is proposed with an increased port height of

1.0m and a vertical discharge angle of 10o

. This modified design, indeed, does not exhibit any bottom attachment as shown in Figure 9-3.

The trajectory predictions of three buoyant jet integral

models (UOUTPLM, UDKHDEN and JF [Jirka-Fong])

and of CORMIX1 (flow class H2) are given in Figure 9-3.

Also shown is the width prediction for CORMIX1. All four

submerged plume trajectories are qualitatively similar;

the deviations among trajectories is contained within the

plume outline (as indicated by CORMIX1) and well

within the usual scatter of experimental data. The TDZ

is again limited (order of 10 m) as predicted by any of

the four models. The jet integral models are, of course,

limited in their applicability to the submerged jet region

before surface interaction. Only CORMIX1 is applicable

to the actual interaction process and the subsequent

buoyant spreading along the water surface. This process is indicated by the width boundary in Figure 9-3.

Figure 9-1. Design case AA: vertical ambient density profile

in typical summer conditions.
Considerable differences exist in the predicted surface

dilution at the point of surface interaction. UOUTPLM

and UDKHDEN predict a flux-averaged dilution of 212

and 495, respectively. On the other hand, JF and

CORMIX1 predict minimum (centerline) dilutions of

220 and 146, respectively. Even if the UDKHDEN

model predictions are divided by a factor of 1.7, in order

to account for the typical ratio of flux-averaged and

minimum dilutions a considerable difference remains

relative to the lower dilution value of CORMIX1. To shed

further light on this disagreement the predictions of the

four models can be compared to what is probably the

Figure 9-3. Case AA2: single port discharge (modified design) in unstratified winter conditions; comparison of jet integral

models and CORMIX1.

Figure 9-2. Case AA1: single port discharge (initial design) exhibiting bottom attachment as predicted by CORMIX1.
most reliable and comprehensive available field data

set on submerged discharges. Lee and Neville-Jones

(1987) report several hundred individual observations

of minimum surface dilution for three single port submerged outfalls for municipal discharges in the United

Kingdom. All of these outfalls are somewhat more

dominated by buoyancy than design case AA2. (This

is indicated, for example, by the fact that CORMIX1

predicts a flow class H1 for these outfalls). The predictions of all four models are compared with the normalized field observations for minimum surface dilution

(Figure 9-4). The solid line presents the best-fit regression line for all data points. The average dilution given

by both USEPA models is a factor of 4 (300%) larger

than the observed minimum dilution. When the dilution

predictions are converted to minimum dilutions (factor

1.7) the overprediction is still by about 130%. The JF

model overprediction is about 50%. CORMIX1, on the

other hand, lies within about 15% with the observations. (Note that the model coefficients of CORMIX1

have been chosen through extensive comparison with

basic laboratory data, so that this good agreement

presents indeed a model validation and not some

forced best-fit). On the basis of this comparison it may

be concluded that the jet integral models (notably

UOUTPLM and UDKHDEN) are quite non-conservative and tend to overestimate actual plume dilutions, at

least for unstratified ambients. The prediction disagreement for Case AA2 (Figure 9-3) may be considered in light of this conclusion.

The legal mixing zone LMZ (25% width) is not attained

in the hydrodynamic near-field but rather in the far-field

as shown by the CORMIX1 predictions of Figure 9-5. In

fact, the LMZ is reached at a downstream distance of

about 600 m when the surface plume is in the buoyant

spreading regime. At this point, the average dilution has

increased to about 250 and the plume half-width is about

75 m with a plume thickness of 1.7 m. Actual plume

interaction with the bank takes place at a further downstream distance of about 760 m. This result illustrates

the practical fact that legal mixing zone definitions can

often imply sufficiently large distances which then include far-field mixing processes. Simple jet integral

models do not address this aspect, while CORMIX1 has

been implemented to deal with such generalities.

9.2.4 Case AA3: Modified Design, Summer

Conditions

The drastic effect of ambient stratification on plume

near-field behavior is shown in Figure 9-6. With any of

the four predictive models the plume is predicted to

reach its terminal level of about 3 to 5 m above the

bottom at a distance of about 10 m downstream. The

differences among the predicted trajectories are small.

The TDZ is reached about 8 m downstream as indicated

by CORMIX1 (flow class S3). The predicted dilution

values at the terminal level show, again, more variability.

If minimum terminal dilutions are compared, then

UOUTPLM, = 16/1.7 = 9, CORMIX1, = 16, and

UDKHDEN, = 26/1.7 = 15, provide lower-end (conservative) predictions, while JF, = 26, is somewhat

higher.

The CORMIX1 predictions in Figure 9-6 also show the

formation of the internal stratified layer (initial thickness

2.4 m) and its gradual collapse and widening with additional mixing. The full development in the far-field is

illustrated again in Figure 9-5. The behavior under stratified summer conditions is in marked contrast with the

unstratified winter conditions (Case AA2). The difference in dilution is notable (related to the much shorter

buoyant jet trajectory in the near-field) as is the much

thinner internal layer. The LMZ is reached at about 680

m where the plume half-width is about 75 m and the

plume thickness about 0.3 m.

9.3 Case BB -Multiport Diffuser: Municipal

Sewage Discharge into Coastal Bay

9.3.1 Ambient and Discharge Conditions

A multiport diffuser is used for the discharge of treated

sewage water from a municipality located on a bay. The

Figure 9-4. Comparison of observed minimum surface

dilution for three submerged single port outfalls

(Lee and Neville-Jones, 1987) with predictions of

jet integral models and CORMIX1.
proposed diffuser location is 10 km offshore with an

ambient water depth of 30 m. In a preliminary evaluation two ambient design cases are to be investigated;

1) A weakly stratified ambient with a density variation

from 1,023.2 kg/m3 at the surface to 1,026.4 kg/m3 at the

bottom. Figure 9-7 shows the actual density variation, together with the schematizations adopted for different models. 2) A uniform ambient with a density of

Figure 9-6. Case AA3 : single port discharge (modified design) in stratified summer conditions; comparison of jet integral

models and CORMIX1.

Figure 9-5. Cases AA2 and AA3: predicted (CORMIX1) far-field behavior for single port discharge (modified design) in winter

and summer conditions.
1,026.0 kg/m3

. In both cases the ambient design velocity is 0.156 m/s for the prevailing coastal current.

The discharge flow rate is 20 m3

/s (460 MGD) with a

freshwater density of 998.0 kg/m3

.

The preliminary design calls for a total diffuser length

of 2000 m with a perpendicular alignment relative to

the prevailing current direction. The diffuser employs

80 vertical risers with 8 ports attached per riser and

discharging in a circular fashion. The port diameter is

0.14 m, the port height is 1.5 m above bottom and the

port angle is 0° (i.e. horizontal).

The legal mixing zone (LMZ) is prescribed by a distance

of 30 m extending in any direction from the diffuser line.

No toxic substances are included in this discharge.

9.3.2 Case BB1: Stratified Ambient

When applying any model to a complex diffuser geometry with riser/port assemblies, some model simplification

is needed. In case of the USEPA multiport models

(UDKHDEN,UMERGE and ULINE) the user must, in

fact, substitute a series of single ports equally spaced

along the diffuser line (thus, in this present case 80 x 8

= 640 ports). On the other hand, the input element of

CORMIX2 collects all the pertinent information about the

riser/port assemblies, the system then concludes that

the net horizontal momentum flux for this diffuser is zero

and treats the diffuser as an alternating diffuser with a

vertical equivalent slot discharge. Thus, in either case,

the local details of the eight individual buoyant jets

discharging from each assembly are neglected.

Figure 9-8 summarizes the predictions of the jet models

UDKHDEN and ULINE and of the expert system CORMIX2 (flow class MS5). All three models indicate a

terminal layer at about 10 m above the bottom varying

between 8 m and 12 m. Also all three models show

limited variability for the predicted average dilution at the

terminal level, _ , which is 137 for ULINE, 212 for

UKHDEN, and 166 x 1.4 = 232 for CORMIX2, using an

average/minimum dilution factor of 1.4 for two-dimensional buoyant jets. All these dilution values

may be scrutinized as to whether the mixed effluent flow

per unit diffuser length, _ ⁄ , exceeds the available

ambient approach flow, , for the layer between

bottom and terminal level. Denoting the ratio

= (_ ⁄ )⁄( ) one finds =0.7 for ULINE, =1.7

for UDKHDEN, and =1.4 for CORMIX2. In a strictly

Figure 9-7. Design case BB: vertical ambient density profile

for design conditions.

Figure 9-8. Case BB1: multiport diffuser discharge under stratified conditions; comparison of jet integral models and CORMIX2.
two-dimensional flow (i.e. if a diffuser section or the

entire diffuser length were bounded by lateral walls)

any value > 1 is not possible in steady-state. However, for the actual three-dimensional diffuser the diffuser entrainment demand can also be met by lateral

flow toward the diffuser line. Futhermore, additional

freedom to entrain water exists for the internally

trapped plume ( < where is the water depth).

Also, note that for low ambient velocity conditions

→0) the above test becomes unreliable for evaluating model performance. Thus, for the present case

of an internally trapped plume from a three-dimensional diffuser all three model predictions appear reasonable.

Note that trajectory information is provided by

UDKHDEN and CORMIX2 while ULINE does not provide any spatial data on plume behavior. The LMZ is

predicted by CORMIX2 to have a minimum dilution of

116.

At the transition to the far-field CORMIX2 indicates an

initial internal layer thickness of about 16 m. As shown

in the far-field plan view of Figure 9-9 this internal layer

is gradually spreading, decreasing in thickness, and

experiencing a slight additional mixing in the buoyant

spreading phase. Thus, at 10 km downstream from the

diffuser line the average dilution is 313, with a halfwidth of the effluent field of 4.2 km and a thickness of

4.7 m.

9.3.3 Case BB2: Uniform Ambient

The corresponding model predictions for the unstratified

case are given in Figure 9-10. CORMIX2 indicates a flow

class MU8 which includes a vertically fully mixed nearfield with an average dilution, _

= 512. Although its model

printout does not specifically state so, ULINE also predicts a vertically mixed flow with a lower dilution _

= 368.

In contrast, UDKHDEN does not recognize the destabilizing effect of the vertically limited environment in crossflow and predicts a plume with a high surface dilution _

= 835 and with width dimensions that are of the order of

the water depth (Figure 9-10). Defining the ratios

=( ⁄ )⁄( ) one finds =0.8 for ULINE, =1.0

for CORMIX2 and = 1.8 for UDKHDEN. The latter

result, together with the fact that the model — while

predicting plume dimensions of the order of the water

depth — does not address the constraint of the limited

ambient depth, indicates that UDKHDEN is not applicable in this case. More generally, it appears that

UDKHDEN is an unreliable model for most multiport

diffuser applications in unstratified ambients. The same

reservation would hold for the model UMERGE (not

plotted here). ULINE indicates slightly more conservative dilution values than CORMIX2. It may be overly

conservative, however, since the ULINE model coefficients are based on a single set of experiments by

Roberts (1977) which did not include the additional

mixing effect of the high velocity discharge jets as is

common in actual diffuser installations (this has been

pointed out in a discussion by Jirka, 1979).

The far-field behavior of the diffuser plume is plotted in

Figure 9-9. While the plume is fully mixed in the nearFigure 9-9. Cases BB1 and BB2: predicted (CORMIX2) far-field behavior for multiport diffuser plume in stratified and uniform

conditions.
field (about 100 m downstream; see Figure 9-10) it

restratifies and lifts off from the bottom. At a distance

of 1.5 km downstream the vertical thickness has reduced to 22.0 m (compared to the initial thickness of

30.0 m equal to the water depth). At 10 km downstream

the plume has thinned to about 11.3 m while spreading

to a half-width of 4.0 km and attaining an average

dilution of 703.

9.4 Case CC -Single Port Discharge: Brine

Discharge From an Oil Field

9.4.1 Ambient and Discharge Conditions

Brine from drilling operations in a coastal oil field is to

be discharged into coastal waters. The proposed discharge site is 250 m offshore at a local water depth of

20 m. The ocean water is weakly stratified with a

pycnocline at 15 m above the bottom. However, because of the strongly negatively buoyant brine discharge the density distribution above the pycnocline

appears unimportant and the ambient can, in fact, be

assumed at a constant density of 1,025.0 kg/m3 corresponding to the lower layer density. Ambient design

velocities range from 0.1 m/s to 0.25 m/s. The bottom

is sandy with a Darcy-Weisbach friction factor of 0.015.

The brine flow rate is 0.03 m3

/s with a density of

1,070.0 kg/m3

, thus much heavier than the ocean

water. The effluent contains several toxic metals, including copper at a concentration of 380 ppb. The

extent of the LMZ corresponds to the water depth of

20 m. The TDZ is governed by a CMC concentration

for copper of 40 ppb.

The initial design proposes a low velocity discharge

(3.8 m/s) with a port of 0.1 m diameter at a 2.0 m height

above the bottom, angled at 60° above horizontal and

pointing laterally across the cross-flow (cross-flowing

discharge).

9.4.2 Case CC1: Low Discharge Velocity

Design, Weak Current

Even though, in principle, they ought to be applicable

for negatively buoyant discharges the USEPA models

(UOUTPLM and UDKHDEN) do not provide any predictions for this cross-flowing upwardly angled discharge with a complex three-dimensional trajectory.

Predictions are limited to CORMIX1 and the JirkaFong (JF) integral model. The near-field plume configuration for the two model predictions is shown in

Figure 9-11a with (i) longitudinal and (ii) transverse

side views, respectively. CORMIX1 predicts a flow

class NV2 with buoyant upstream intrusion along the

bottom after impingement of the falling jet. The two

buoyant jet trajectories (JF and CORMIX2) are in

reasonably good agreement prior to impingement. The

predicted minimum dilution is lower for CORMIX1

( = 22) than for JF (56). The extent of the upstream

intrusion is of the order of 20 m from the impingement

point. A thin bottom layer of about 0.5 m thickness is

formed and spreads laterally as the bottom plume is

advected downstream. The TDZ is very short, of the

order of 1 m from the efflux point. The conditions at the

LMZ (not shown in Figure 9-11a) indicate a thin layer

of 0.35 m thickness, 18.0 m half-width with an average

dilution of 40.

9.4.3 Case CC2: Low Discharge Velocity

Design, Strong Current

When the ambient current increases from 0.1 m/s to

0.25 m/s the downstream buoyant jet deflection is

accentuated while the upstream buoyant extension is

Figure 9-10. Case BB2: multiport diffuser discharge in uniform ambient; comparison of jet integral models and CORMIX2.
minimized. Figure 9-11b shows CORMIX1 (flow class

NV2) and JF predictions. The discrepancy between

predicted minimum dilutions is further increased

( = 22 versus 140). Such complex three-dimensional trajectories represent some of the most severe

tests for model application, and in the absence of

detailed experimental data for such phenomena it is

difficult to favor one model over another.

The upstream intrusion along the bottom is minimal in

the present case (order of 2 m) and the bottom density

current is thicker and less wide. At the LMZ distance

the plume half-width is only 8.0 m with a thickness of

0.60 m and an average dilution of 45.

9.4.4. Case CC3: High Discharge Velocity

Design, Strong Current

In order to maximize near-field dilution a high exit

velocity design (15.2 m/s) is evaluated by halving the

port diameter to 0.05 m. The results are shown in

Figure 9-12. When compared to Figure 9-11b, this

shows the significant effect of increased jet diffusion in

the near-field. The buoyant jet shows much more rapid

mixing, and, consequently, is more liable to advection

by the ambient current. CORMIX1 (flow class NV1) no

longer predicts an upstream intrusion after the more

gradual bottom approach. There are differences in the

predicted jet trajectories, as far as maximum height of

rise and bottom approach are concerned. At the LMZ

these buoyant jets are predicted to be in the water

column without any bottom contact yet. The minimum

dilution values are = 247 for JF and 119 for

CORMIX1, respectively. The comparison between

Figure 9-11b and 9-12 illustrates how LMZ constraints

sometimes are met in the hydrodynamic near-field and

at other times in the far-field, depending on the interplay of ambient and discharge conditions.

9.5 Case DD Multiport Diffusers: Cooling

Water Discharge into Shallow Sound

9.5.1 Ambient and Discharge Conditions

A once-through cooling water system for a thermalelectric power plant discharges the heated cooling

Figure 9-11. Cases CC1 and CC2: negatively buoyant discharge from single port; low exit velocity design under a) weak and b)

stronger ambient current.
water through a submerged multiport diffuser. At a

distance of 500 m offshore, a shallow relatively flat

area exists with an ambient water depth of 10.3 m.

The water is unstratified with an average temperature

of 20°C and ocean salinity. The velocity field is tidal

ranging from slack tide (0.0 m/s) to weak velocities

(about 0.1 m/s) to a maximum velocity (0.5 m/s). The

cooling water flow rate is 67 m3

/s with a discharge

temperature rise of 20.5°C above ambient and the

same salinity.

A staged diffuser design of 260 m length is proposed

with a perpendicular alignment relative to the tidal

currents. The diffuser consists of 32 ports with a port

height of 0.5 m, port diameter of 0.6 m and a vertical

angle of 20° above horizontal.

No LMZ is specified. Rather, the predictive results are

to be interpreted so as to make an LMZ proposal to the

state regulatory authority.

9.5.2 Case DD1: Weak Tidal Current

None of the USEPA diffuser models are applicable for

such shallow water diffusers with strong momentum

flux and unstable near-field mixing. If they were used,

UOUTPLM and UDKHDEN would predict vertical

plume width far in excess of the available water depth.

ULINE, on the other hand, is limited to pure plume

discharges without any directed discharge momentum

flux.

Thus, reliable predictions are limited to CORMIX2 as

shown in the plan view of Figure 9-13. For this case of

a weak current, CORMIX2 (flow class MU5) indicates

an initially, vertically fully mixed diffuser plume. The

plume gets gradually deflected by the weak crossflow

and begins to re-stratify (lift off the bottom) after a

distance. Gradual, lateral spreading and vertical thinning of the diffuser plume takes place. The induced

temperature rise is 2.7°C in the near-field and drops to

1.0°C at a distance of about 1500 m. (Any potential

heat loss to the atmosphere is neglected in these

conservative mixing predictions).

Figure 9-12. Case CC3: negatively buoyant discharge from single port; high exit velocity design with strong ambient current.
Figure 9-13 illustrates vividly the strong effect of the

directed momentum flux from shallow multiport diffusers and the ability to induce currents over considerable distances.

9.5.3 Case DD2: Slack Tide

Stagnant ambient conditions always represent a limiting case for any mixing analysis. Since there is no

ambient advective mechanism they are always associated with an unsteady flow field and mixing process,

including potential large scale recirculation effects.

The CORMIX2 (flow class MU5) predictions are given

in Figure 9-14 for unsteady conditions. The plume is

now undeflected, but has similar mixing characteristics

as the slightly deflected plume of Case DD1. However,

at some distance (about 680 m) the predictions are

terminated since the induced plume velocities have

become negligibly small so that a transient recirculating flow would be set up. Corresponding messages are

printed out by the expert system along with the advice

to conduct predictions for stagnant ambients only as a

special limiting condition.

9.5.4 Case DD3: Strong Tidal Current

The effect of a strong tidal current (0.5 m/s) is to

generate a strongly deflected diffuser plume (Figure

9-15) as predicted by CORMIX2 (flow class MU6). A

rapid deflection and greatly increased mixing take

place within the diffuser vicinity. The re-stratifying

plume is then advected by the ambient current and

grows in width and diminishes in vertical thickness, in

form of a surface buoyant spreading process.

Figure 9-14. Case DD2: staged multiport diffuser for cooling

water discharge; CORMIX2 predictions for slack

tidal conditions.

Figure 9-15. Case DD3: staged multiport diffuser for cooling

water discharge; CORMIX2 predictions for

strong tidal current.

Figure 9-13. Case DD1: staged multiport diffuser for cooling

water discharge; CORMIX2 predictions for weak

tidal current.
In summary, the great variability among diffuser plume

patterns (Figures 9-13, 9-14, and 9-15) suggests that

a complete assessment of initial mixing processes

should, indeed, include the whole spectrum of ambient

conditions. It is often difficult to define a single “typical”

design condition for mixing analysis. On the other

hand, a rapid evaluation of several ambient conditions

and of alternative designs is readily possible within the

framework of presently available models.

9.6 References

Doneker, R.L., and G.H. Jirka. 1990. CORMIX1: An

Expert System for Hydrodynamic Mixing Zone Analysis of Conventional and Toxic Submerged Single Port

Discharges. Technical Report, U.S. EPA, Environmental Research Laboratory, Athens, GA.

Jirka, G.H. 1979. Discussion of “Line Plume and

Ocean Outfall Dispersion” by P.J.W. Roberts, J. of the

Hydraulics Div., ASCE, Vol. 105, HY 12.

Lee, J.H.W. and P. Neville-Jones. 1987. Initial Dilution

of Horizontal Jet in Crossflow. J. Hydraulic Engrg.,

ASCE, Vol. 113, No. 5.

Muellenhoff, W.P., et al. 1985. Initial Mixing Characteristics of Municipal Ocean Discharges (Vol. 1&2).

U.S.E.P.A., Environmental Research Laboratory, Narragansett, R.I.

Roberts, P.J.W. 1977. Dispersion of Buoyant Waste

Discharge from Outfall Diffusers of Finite Length. Rep.

No. KH-R-35. W.M. Keck Lab. of Hydraulics and Water

Resources, California Institute of Technology,

Pasadena, CA, 183 pp.
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