Chapter 5 – Environmental Hydraulics

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5.1 INTRODUCTION

The thermal, chemical, and biologic quality of water in rivers, lakes, reservoirs, and nearcoastal areas is inseparable from a consideration of hydraulic engineering principles;

therefore, the term environmental hydraulics. In this chapter we discuss the basic princi-ples of water and thermal budgets as well as mixing and dispersion.

5.2 WATER AND THERMAL BUDGETS

5.2.1 Water Budget

A water budgetis a statement of the law of conservation of mass or

(change in storage) (input) (output) (5.1)

and the expressions of the water budget can range from simple to very complex. For exam-ple, consider the lake or reservoir shown in Figure 5.1. For this situation, a generic waterbudget could be written as follows:

d

d

S

ts (Ic Io Ig Pr Rr) (Ev Tr Gs Oc W) (5.2)

CHAPTER 5

ENVIRONMENTALHYDRAULICS

Richard H. FrenchWater Resources Center

Desert Research Institute

University and Community College System of Nevada

Reno, Nevada

Steven C. McCutcheonEcosystems Research Division

National Exposure Research Laboratory

U.S. Environmental Protection Agency

Athens, Georgia

James L. Martin

AScI CorporationAthens, Georgia

5.1

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Source: HYDRAULIC DESIGN HANDBOOK


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where Ic channel inflow rate, Io overland inflow rate, Ig groundwater inflow rate,Pr precipitation rate, Rr return flow rate, Ev evaporation rate, Tr transpirationrate, Gs groundwater seepage rate, Oc channel outflow rate, W consumptive with-drawal, and S

s

lake/reservoir storage rate at time t(volume).The solution of Eq (5.2) quantifies the terms, and, in many cases, the goal of the mod-

eling effort is to estimate the value of a single term or group of terms: for example, evap-otranspiration (Ev Tr). The reliability of using a water budget is directly related to theaccuracy of the prediction techniques used, the availability and quality of gauged data, andthe time period involved. Among the methods of evaluating the individual terms in Eq.(5.2) are the following:

Channel inflow and outflow (Ic and Oc )gauging, statistical simulation.

Overland inflow (Io)gauging, rainfall-runoff relationships.

Groundwater inflow and seepage rate (Ig and Gs)seepage equations, gauging. Precipitation (Pr)gauging, statistical simulation (Smith, 1993).

Evaporation and transpiration (Eand T)gauging, evaporation/transpiration predic-tion relationships (Bowie et al. 1985; Shuttleworth, 1993).

Return flow and withdrawal (Rr and W)gauging.

5.2 Chapter Five

FIGURE 5.1 A hypothetical lake illustrating the variables in the water budget.



ENVIRONMENTAL HYDRAULICS


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5.2.2 Thermal Budget

The total thermal budgetfor a body of water includes atmospheric heat exchange at the

air water interface (usually the dominant process), the effects of inflows (tributaries,wastewater, and cooling water discharges), heat resulting from chemical-biological reac-tions, and heat exchange with the stream bed. In the following sections, the primary com-ponents of the air-water interface heat budget will be briefly discussed; for further detailsthe reader is referred to Bowie et al., (1985), McCutcheon (1989), or Shuttleworth(1993).

Atmospheric heat exchange at the air-water interface is given by

H Qs Qsr Qa Qar Qbr Qe Qc (5.3)

where H net surface heat flux, Qs shortwave radiation incident to the water surface

[3300 (kcal/m2)/h], Qsr reflected shortwave radiation [525 (kcal m2)/h], Qa incom-

ing longwave radiation from the atmosphere (225360 kcal/m2/h), Qar reflected long-wave radiation [515 (kcal m2)/hr], Qbr longwave back radiation emitted by the waterbody [220345 (kcal m2)/h], Qe energy utilized by evaporation [25900 (kcal m

2)/h],and Qc energy convected to or from the body of water (3550 kcal m

2/hr). Note thatthe ranges given are typical for the middle latitudes of the United States (Bowie et al.,1985).

The equations for estimating the terms of the thermal budgets use a mixed set of units,and appropriate conversions among the different units used are provided in Table 5.1.

5.2.2.1 Net atmospheric shortwave radiation (Qs Qsr) The net shortwave radiation(Qsn) is that portion of the incident shortwave radiation captured at the ground, taking intoaccount losses caused by reflection. Although solar radiation can be measured with spe-cialized meteorological stations equipped with radiometers, these instruments requirepainstaking calibration and maintenance. In most cases, measured values of solar radia-tion are not available at the location of interest and must be estimated from equations.Among the formulations for estimating net shortwave solar radiation is

Qsn Qs Qsr 0.94Qsc(1 0.65C2

c) (5.4)

where Qsc clear sky solar radiation [kcal m2)/h) and Cc fraction of sky covered by

clouds (Anderson, 1954; Ryan and Harleman, 1973). It is pertinent to note that Eq. (5.4)

Environmental Hydraulics 5.3

TABLE 5.1 Useful Energy Conversions for Energy Budget Calculations

1Btuft2/day = 0.131 W/m2 = 0.271 Ly/day = 0.113 (kcal m2)/h

1 watt/m2 = 7.61 Btu ft2)/day = 2.07 Ly/day = 0.86 (kcal m2)/h

1 Ly/day = 0.483 W/m2 = 3.69 (Btu/ ft2)/day = 0.42 (kcal m2)/h

1 (kcal m2)/hr = 1.16 W/m2 = 2.40 Ly/day = 8.85 (Btu ft2)/day

1 kpa = 10 mb = 7.69 mm Hg = 0.303 in (Hg)

1 mb = 0.1 kpa = 0.769 mm Hg = 0.03 in (Hg)

1 mm Hg = 1.3 mb = 0.13 kpa = 0.039 in (Hg)

1 in Hg = 33.0 mb = 25.4 mm Hg = 3.3 kpa

Abbreviations Ly Langleys; mb millibar; and Btu British Thermal Unit





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5.4 Chapter Five

assumes average reflectance at the Waters surface and uses clear sky solar radiation. Insome situations, the effects of atmospheric attenuation are much greater than normal andmore complex equations are required (e.g., 1972). Clear sky radiation (Qsc) can be esti-

mated as a function of calendar month and latitude from Fig. 5.2.Shortwave solar radiation is absorbed at the waters surface and penetrates the water

column, depending on the wavelength of the radiation, the properties of the water, and thematter suspended in the water. The degree of penetration of shortwave solar radiation(sunlight) into the water column has a significant effect not only on water temperature butalso on the rate of photosynthesis by aquatic plants and the general clarity, color and aes-thetic quality of the water. Thepenetration of shortwave solar radiation is described by

IIoexp (ke y) (5.5)

where I light intensity at depthy, Ke extinction coefficient, andIo light intensityat the surface (y 0).

Values of the extinction coefficient can be estimated by several methods. For example,measurement of total light penetration into a water column can be made by using a pyre-heliometer positioned at the surface that measures the total incoming solar radiation.Simultaneously, an underwater photometer is lowered and the radiation is recorded at eachof a series of depths throughout the water column. Then, a value ofKe can be estimatedby linear leastsquares regression. An alternative but traditional, simpler, and less accu-rate method to estimate Ke is to lower a target into the water column until, by eye, the tar-get just disappears. A standardized target (Secchi disk) is commonly used, and a numberof investigators (Beeton, 1958; French et al., 1982; Sverdrup et al, 1942;) have developed

empirical relationships between, the Secchi disk depth (ys) and the extinction coefficientof the form.

Ke (1.2 to

ys

1.9) (5.6)

Finally, the depth (ye) at which 1 percent of the surface radiation still remains (theeuphotic depth) is given from Eq. (5.5) as

ye 4

K

.6

e

1 (5.7)

5.2.2.2 Net atmospheric long-wave radiation (Qa Qar) Atmospheric radiation is char-acterized by much longer wavelengths than solar radiation because the major emittingelements are water vapor, carbon dioxide, and ozone. The approach generally used toestimate this flux involves the empirical estimation of an overall atmospheric emissivityand the use of the Stephan-Boltzman law (Ryan and Harleman, 1973). Swinbank (1963)developed the following equation, which has been used in many water quality models:

Qan Qa Qar 1.16 1013

(1 0.17C2

c)(Ta 460)6

(5.8)

where Qan net longwave atmospheric radiation (Btu ft2/day), Cc fraction of sky cov-

ered by clouds, and Ta

dry bulb air temperature (F).

5.2.2.3 Long-wave back radiation (Qbr) The long-wave back radiation from a watersurface in most cases is the largest of all the fluxes in the heat budget (Ryanand Harleman, 1973). The emissivity of a water surface is well known; therefore, thisflux can be estimated with a high degree of accuracy as a function of the water surfacetemperature:





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FIGURE5.2

Clearskysolarradia

tion.(FromHamonetal.1954)





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Qbr 0.97T4

s (5.9)

where Qbr longwave back radiation (cal/m2/s), Ts surface water temperature (

0K),

and Stefan-Boltzman constant (1.357 108 cal m2/s/K4)

5.2.2.4 Evaporative heat flux (Qe) Evaporative heat loss (kcal/m2/s) occurs as a result

of the change of state of water from a liquid form to vapor and is estimated by

Qa LwEv (5.10)

where Lw latent heat of vaporization ( 597 0.57Ts, kcal/kg), Ts surface water tem-perature (C), Ev evaporation rate (m/s), and water density (kg/m3).

A standard expression for evaporation from a natural water surface is

Ev (a bW)(es ea) (5.11)

where Ev evaporation rate (m/s), a and b empirical coefficients, W wind speed atsome specified distance above the water surface (m/s), es saturation vapor pressure atthe temperature of the water surface (mb), and ea vapor pressure of the overlying atmos-phere (mb). In many cases, the empirical coefficient a has been taken as zero with 1 109 b 5 109 (Bowie et al., 1985). The saturated vapor pressure can be estimated(Thackston, 1974) by

es

exp17.62

Ts

9

50

4

1

60

(5.12)

where es is in inches of Hg, and Ts water surface temperature (F). There are a numberof ways of estimating ea, depending on the available data. For example, if the relativehumidity (RH) is known, then

RH ee

a

s (5.13)

and then if the wet bulb temperature and atmospheric pressure are known (Brown andBarnwell, 1987)

ea es 0.000367Pa(Ta Twb)

1

Tw1b

5

71

32

(5.14)

where all pressures are in (in Hg), all temperatures are in (F), Pa atmospheric pressure,and Twb wet bulb temperature. The relationship among the air and wet bulb tempera-tures (F) and relative humidity (Thackston, 1974) is

Twb (0.655 0.36RH)Ta (5.15)

There are many equations for estimating the rate of evaporation. For example, Jobson(1980) developed a modified formula that was used in the temperature modeling of the

San Diego Aqueduct and subsequently was modified for use on the Chattahoochee Riverin Georgia (Jobson and Keefer, 1979). McCutcheon (1982) noted that, in many models,the wind speed function is a catchall term that compensates for many factors, such as (1)numerical dispersion in some models, (2) the effects of wind direction, fetch, channelwidth, sinuosity, bank, and tree height, (3) the effects of depth, turbulence, and lateralvelocity distribution; and (4) the stability of air moving over the stream. (Fetch is the dis-tance over which the wind blows or causes shear over the waters surface.) Finally, it is

5.6 Chapter Five





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important to note that evaporation estimators that work well for lakes or reservoirs will notnecessarily provide the same level of performance when used in streams, rivers, or con-structed open channels.

5.2.2.5 Convective heat flux (Qc). Convective heatis transferred between air and waterby conduction and is transported to or from the air-water interface by convection. The con-vective heat flux is related to the evaporative heat flux(Qe) by theBowen ratio (Bowie etal., 1985), or

RB Q

Qc

e

(6.19 104)Pa Tess

T

ea

a (5.16)

where all temperatures are in (C), all pressures are in (mb), andRB Bowen ratio.

5.2.2.6 Conclusion. The foregoing is a brief summary of the approaches used most fre-quently to estimate surface heat exchange in numerical models. The reader is referred toother publications for a more detailed discussion of the approaches (Bowie et al., 1985)and meteorological data requirements (Shanahan, 1984). Note that each situation shouldbe considered carefully from the viewpoint of specific factors that must be taken intoaccount. For example, in most lakes, estuaries, and deep rivers, the thermal flux throughthe bottom is not significant. However, in water bodies with depths less than 3 m (10 ft),bed conduction of heat can be significant in determining the diurnal variation of temper-atures within the body of water (Jobson, 1980, Jobson and Keefer, 1979).

5.3 EFFECTS AND CAUSES OF STRATIFICATION

5.3.1 Effects

The density of water is strongly affected by temperature and the concentrations of dis-solved and suspended solids. Regardless of the cause of differences in water density, waterwith the greatest density is found at the bottom, whereas water with the least density residesat the surface. When density gradients are strong, vertical mixing is inhibited. Stratification

is the establishment of distinct layers of water of different densities (Mills et al., 1982).Stratification is enhanced by quiescent conditions and is destroyed by in a body of water-phenomenasc that encourage mixing (wind stress, turbulence caused by large inflows, anddestabilizing changes in water temperature). In many bodies of water (rivers, lakes, andreservoirs), stratification is the single most important phenomena affecting water quality.

When stratification is absent, the water column is mixed vertically and dissolved oxy-gen (DO) is present in the vertical water column from the top to the bottom: that is, fullymixed water columns do not have DO deficit problems. For example, when stratificationoccurs, in reservoirs and lakes mixing is limited to the epliminion or surface layer. Sincestratification inhibits, vertical mixing is inhibited by stratification, and reaeration of the

bottom layer (the hypoliminion) is inhibited if not eliminated. The thermocline (the layerof steep thermal gradient between the epiliminion and hypoliminion) limits not only mix-ing but also photosynthetic activity as well. The hypolimnion has a base oxygen demandand benthic matter and the settling of particulate matter, from the epiliminion only addsto this demand. Therefore, while the demands of DO in the hypoliminion increase duringthe period of stratification, inhibition of mixing between the epiliminion and thehypolimnion and the lack of photosynthetic activity deplete the DO concentrations in the






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5.8 Chapter Five

hypolimnion. Finally, a rule of thumb suggests that when water temperature is the pre-dominant cause of differences in water density a temperature gradient of at least 1C/m isrequired to define the thermocline (Mills et al., 1982).

The density of water can be estimated by

T s (5.17)

where water density (kg/m3), T water density as a function of temperature, ands increments in density caused by solids.

5.3.2 Water Density as a Function of Temperature

A number of formulations have been proposed to estimate T and among these areT 999.8452594 6.793952 10 2 Te

9.095290 103 Te2 1.001685 104 Te3 (5.18)

1.120083 106 Te4 6.536332 109 Te5

where Te water temperature in C (Gill, 1982).

5.3.3 Water Density as a Function of Dissolved Solids or Salinity andSuspended Solids

In most cases, data for dissolved solids are in the form of total dissolved solids(TDS); however, in some cases, salinity may be specified. The density increment for dis-solved solids can be estimated by

TDS CTDS(8.221 104

3.87 106

Te 4.99 108 Te2) (5.19)

(Ford and Johnson, 1983), where CTDS concentration of TDS (g/m3 or mg/L). If the con-

centration of TDS is specified in terms of salinity (Gill, 1982).

SL CSL(0.824493 4.0899 103

Te 7.6438 105

Te2

8.2467 107

Te3

5.3875 109

Te4)

CSL1.5

(5.72466 103

1.0277 104

Te

1.6546 106

Te2) 4.8314 10

4CSL

2(5.20)

where CSL concentration of salinity (kg/m3). The density increment for suspended

solids is

ss Css

1.

S

1

G

103

(5.21)

where SG specific gravity of the suspended sediment. (Ford and Johnson, 1983).The total density increment caused by solids is then

s (TDS or SL) SS (5.22)





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5.4 MIXING AND DISPERSION IN OPEN CHANNELS

Turbulent diffusion (mixing) refers to the random scattering of particles in a flow by tur-

bulent motions, whereas dispersion is the scattering of particles by the combined effectsof shear and transverse turbulent diffusion. Shearis the advection of a fluid at differentvelocities at different positions within the flow.

When a tracer is injected into a homogeneous channel flow, the advective transportprocess can be viewed as composed of three stages. In the first stage, the tracer is diluted bythe flow in the channel because of its initial momentum. In the second stage, the tracer ismixed throughout the cross section by turbulent transport processes. In the third stage, lon-gitudinal dispersion tends to erase longitudinal variations in the tracer concentration. Insome cases, the second stage is eliminated because the tracer discharge has a significantamount of initial momentum associated with it; however, in many cases, the tracer flow is

small and the momentum associated with it is insignificant. In the latter case, the first trans-port stage is eliminated. In this treatment, only the second and third transport stages will betreated, with the implied assumption that if there is a first stage, it can be treated separately.

The reader is cautioned that, in this chapter, y is the vertical coordinate direction andzis the transverse coordinate direction.

5.4.1 Vertical Turbulent Diffusion

To develop a quantitative expression for the vertical turbulent diffusion coefficient,consider a relatively shallow flow in a wide rectangular channel. It can be shown that thevertical transport of momentum in such a flow is given by

v d

d

v

y (5.23)

where shear stress at a distancey above the bottom boundary, fluid density, v vertical turbulent diffusion coefficient, and v longitudinal velocity (French, 1985).Because the one-dimensional vertical velocity profile and shear distribution are known, itcan be shown that

v kv*yd

y

y

d

1

y

y

d

(5.24)

where k von Karmans turbulence constant (0.41), yd depth of flow, v* shear veloc-ity ( gydS), and S longitudinal channel slope (French, 1985). The depthaveragedvalue ofv is

v 0.067ydv* (5.25)

When the fluid is stably stratified, mixing in the vertical direction is inhibited, and oneoften quoted formula expressing the relationship between the unstratified and stratifiedvertical mixing coefficient was provided by Munk and Anderson (1948):

vs 1 3.3

v

3 Ri)1.5 (5.26)

where vs the stratified vertical mixing coefficient.





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5.4.2 Transverse Turbulent Diffusion

In the infinitely wide channel hypothesized to derive Eq. (5.24), there is no transverse

velocity profile; therefore, a quantitative expression for t , the transverse turbulent diffu-sion coefficient, cannot be derived from theory. The following equations to estimate tderived from experiments by Fischer et al., (1979), and Lau and Krishnappen (1977).In straight rectangular channels, an approximate average of the results available is

t 0.15ydv* 50% (5.27)

where the 50 percent indicates the error incurred in estimating t. In natural channels, tis significantly greater than the value estimated by Eq. (5.27). For channels that can beclassified as slowly meandering with only moderate boundary irregularities

t 0.60ydv* 50% (5.28)

If the channel has curves of small radii, rapid changes in channel geometry, or severebank irregularities, then the value oft will be larger than that estimated by Eq. (5.28).For example, in the case of meanders, Fischer (1969) estimated that

t 25V

R

2

2

y

cv

3d

* (5.29)

where a slowly meandering channel is one in which

RT

c

Vv* 2 (5.30)

and Rc radius of the curve.As stated above, the complete advective transport process in a two-dimensional flow

can be conveniently viewed as composed of three stages. In the second stage, the prima-ry transport mechanism is turbulent diffusion, and a comparison of Eqs. (5.25) and (5.27)shows that the rate of transverse mixing is roughly 10 times greater than the rate of verti-cal mixing. Thus, the rate at which a plume of tracer spreads laterally is an order of mag-nitude larger than is the rate of spread in the vertical direction. However, most channelsare much wider than they are deep. In a typical case, it will take approximately 90 times

as long for a plume to spread completely across the channel as it will take to mix in thevertical dimension. Therefore, in most applications, it is appropriate to begin by assumingthat the tracer is uniformly distributed over the vertical.

In a diffusional process in which the tracer is added at a constant mass flow rate (M*)at the center line of a bounded channel (C/z 0 atz 0 and C/z 0 atz T), thedownstream concentration of tracer is given approximately by

CC

41

x'

n exp

(z' 2

4

n

x'

zo')2

exp

(z' 2

4

n

x'

zo')2

(5.31)

where

C' VM

T

*

yd

5.10 Chapter Five





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x' V

x

T

t2

and

z' T

z

A reasonable criterion for the distance required for complete mixing (where the con-centration is within 5 percent of its mean value everywhere in the cross section) from acenter-line discharge is

L 0.1

V

t

T2 (5.32)

If the pollutant is discharged at the side of the channel, the width over which the mix-

ing must take place is twice that for center-line injection, but the boundary conditions areotherwise identical and Eq. (5.32) applies ifTis replaced with 2T.

5.4.3 Longitudinal Dispersion

After a tracer becomes mixed across the cross section, the final stage in the mixing processis the reduction of longitudinal gradients by dispersion. If a conservative tracer is dis-charged at a constant rate into a channel, the flow rate of which also is constant, there isno need to be concerned about dispersion; however, in the case of an accidental release

(spill) of a tracer into a channel or the release is cyclic, dispersion is important. The one-dimensional equation governing longitudinal dispersion is

Ct V

Cx K

2

2Cx S (5.33)

where K the longitudinal dispersion coefficient and S sources or sinks of materials.The initial work in dispersion, beginning with Taylor (1954), assumed a prismatic chan-nel. However, natural streams have bends, sandbars, side pools, in-channel pools, bridgepiers, and other natural and anthropogenic changes, and every irregularity in the channelcontributes to longitudinal dispersion. Some channels may be so irregular that no reason-

able approximation of dispersion is possible: for example, a mountain stream consistingof pools and riffles.Fischer et al. (1979) presented a number of methods of approximating Kin a natural

open channel. Of these, the most practical is

K 0.01

y

1

dv

V

*

2T

2

(5.34)

Equation (5.33) depends on a crude estimate of t and does not reflect the existence ofdead zones in natural channels. However, it does have the advantage of relying only onthe usually available estimates of depth, velocity, width, and surface slope.

With regard to the solution of the dispersion equation, the following observations arepertinent:

1. The longitudinal dispersion analysis is not valid until the end of the initial period,when

x0.4

V

t

T2

(5.35)






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5.12 Chapter Five

2. In the case of a slug of dispersing material (massM), the longitudinal length of thecloud after the initial period can be estimated approximately by

L 4

2K

t

T2

V

x

T

t2

0.07

0.5

(5.36)

and the peak concentration within the dispersing cloud is

Cmax (5.37)

Note that the observed value of the peak concentration will generally be less than this

estimate because some of the material is trapped in dead zones and some of the typicaltracers (Martin and Mc Cutcheon, 1999) sorb onto sediment particles.

5.5 MIXING DISPERSION IN LAKES AND RESERVOIRS

Important factors in the hydraulic design, operation, and analysis of spills in reservoirsand lakes include (1) determining vertical stratification to guide lake monitoring and thedesign withdrawal structures, (2) locating the plunge point or separation point to deter-mine how inflows mix, (3) computing the dilution and mixing of inflows and the time

required to travel through a reservoir or lake, and (4) determining the quality of with-drawals or outflows and effects on the quality of reservoir water. The elevation and flowthrough withdrawal structures at dams are selected to control flooding and achieve cer-tain water-quality targets or standards. The stratification, mixing, and travel of inflowsare determined to design water-intake structures at dams or other locations in lakes, toforecast the habitat and fisheries that a proposed reservoir may support, and to trackchemical spills or flood waters through reservoirs. This section is based onChaps. 8 and 9 in Martin and McCutcheon (1999), which provide a number of samplecalculations.

Many lakes and reservoirs stratify for part of the year into an epilimnion, thermocline,

and hypolimnion illustrated in Fig. 5.3. The depth and thickness of the thermocline or met-alimnion vary with location and time of the year and even time of the day to a limitedextent. The thermocline represents the interface between a well-mixed surface layer, orepilimnion, and the cooler, deeper hypolimnion. In freshwater lakes, the thermocline isdefined by a minimum temperature gradient of 1C/m. When a distinct interface does notexist, the thermocline, epilimnion, and hypolimnion may not be defined. Mixing process-es also are different in riverine, transition, and lacustrine zones (Fig. 5.3). Mixing in theriverine zone is dominated by advection and bottom shear, and turbulence is generally dis-sipated under the same conditions. Seiche, wind mixing, boundary shear, boundary intru-sion, withdrawal shear, internal waves, and dissipation of turbulence generated elsewherecause mixing in the lacustrine zone. Buoyancy resulting from stable stratification stabi-lizes or prevents mixing. In the transition zone, ending at the plunge point or separationpoint, buoyancy begins to balance the advective force of the inflow. There are threesources of energy for mixing: (1) inflows from tributaries, overland runoff, and dis-charges, (2) withdrawal at dams, discharges at control structures, and natural outflows,and (3) wind shear, solar heating and cooling, heat conduction and evaporation, and othermeteorological forces.

M

A4VKx





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FIGURE5.3

Mixingprocessesinzone

soflakesandreservoirs.(Modifie

dfromFischer,etal.1979)





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Shallow lakes and reservoirs that do not stratify are normally analyzed in the samefashion as rivers or as a completely mixed body of water. For a completely mixed system,the residence time (Tin seconds or more typically years) or time for an inflow to travel

through the body of water is simply tr /Q, where is the volume of the lake (m3) andQ is the sum of the inflows or the average reservoir discharge (m3/s).

Freshwater lakes tend to stratify when the mean depth exceeds 10 m and the residencetime exceeds 20 days (Ford and Johnson, 1986). The densimetric or internal Froude num-berFrd(Norton et al., 1968) provides a better indication of the stratification potential of areservoir where

Frd Frp (5.38)

LL the length of the reservoir (m), yavg the its mean depth (m), g gravitational accel-eration (m/s2), the difference in density over the depth for the internal Fr or betweenthe inflow and surface waters of the lake or reservoir at the plunge point or separationpoint (kg /m3), r average density of the lake for the internal Fr or density of the inflow(Turner, 1973) at plunge or separation points (kg/m3), Vo the average velocity of theinflow (m/s), andyo the hydraulic depth or cross-sectional area divided by the top widthof the inflow (m). The Fr at the plunge pointFrp, also defined in Eq. (5.38), will be usedin the next section. For design projections, the dimensionless density gradient/(yavg)normally is taken to be 10-6 m1 (Norton et al., 1968). If Fr >> 1/, the reservoir is expect-

ed to be well mixed. If Fr


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spring or early summer and persists into the fall or early winter, depending on latitude.The surface heats rapidly, becoming less dense than deeper layers and forming stable dif-ferences in vertical density that inhibit vertical mixing until the fall overturn. As stratifi-

cation develops, wind and currents mix the upper layers and tend to deepen the thermo-cline to form the well-mixed epilimnion. Although storms in late spring and summerepisodically lower the thermocline, the thermocline generally rises as solar heatingincreases until midsummer. After later summer cooling begins, the thermocline deepensuntil the fall overturn occurs. The decreased difference in temperature in the fall with thehypolimnion allows more mixing that deepens the epilimnion and thermocline. The vari-able depth of the thermocline at any time is controlled by seasonal climate, the occurrenceof storms, water temperature, water depth, lake bathymetry, the strength of inflow and out-flow current, and other factors covered in more detail by Chapra and Reckhow (1983),Ford and Johnson (1986), Hutchinson (1957), and Wetzel (1983).

The onset of cooler fall conditions causes the epilimnion to lose heat to the atmos-phere. As heat is lost, mixing tends to become more dominant. The overturning or com-plete mixing of the reservoir or lake dominates as the epilimnion and hypolimnionapproach the same temperature. During winter, lakes and reservoirs remain unstratifiedexcept in the higher latitudes where the hypolimnion approaches 4C and the surfaceapproaches 0C. The slight winter stratification of these colder water bodies is the resultof to the usual decrease in water density as temperature decreases from 4 to 0C. Ice covermaximizes and prevents wind mixing and erosion of the mild differencesm in density.Stratification is so mild that a distinct thermocline does not form and the epilimnion andhypolimnion are not well defined. Winter stratification persists until spring warming meltsthe ice and heats the surface layer to the temperature of the hypolimnion (usually 4C)when the spring overturn occurs.

The arrival of spring begins the cycle of heating and stratification anew. A differencein temperature of just a few degrees results in a difference in density sufficient to inhibitor prevent most vertical mixing in lakes and reservoirs. Vertical mixing is inhibited almostcompletely during summer heating because wind and inflows and outflows do not havesufficient energy to erode the differences in density that arise. The wind and energy avail-able from wind and currents cannot overcome the potential energy differences that tend toprevent mixing of the denser hypolimnion and lighter epilimnion. Fresh water flows intoa saline lake cause salinity gradients that have the same damping effect. Density stratifi-cation also is caused by suspended sediments, primarily resulting in sediment-laden

underflows. Martin and McCutcheon (1998) have illustrated the stratification cycle forwarmwater lakes and reservoirs.

Run-of-the-river reservoirs and shallow lakes that are weakly stratified because of highflows or wind mixing, follow only the general stratification trend. Complete mixing mayoccur during the summer stratification period as a result of wind or runoff events, and thethermocline may be difficult to define. Fall overturn occurs earlier in these bodies of waterthan it does in deeper lakes.

5.5.2 Plunge and Separation PointEnd of the Transition

Between Riverine and Lacustrine Conditions

The plunge pointor separation pointmarks the downstream end of the transition zonedefined where buoyancy begins to exceed advective forces. These points move seasonal-ly and, to a limited degree, during the day. Usually distinguished by a line of foam or float-ing debris across the reservoir or lake, the plunge point occurs when a denser inflow divesbelow the lakes surface and continues to flow along the bottom as a density current. Theseparation point occurs when an underflow has the same density of the lake water at a






16/30

given depth, and separates from the bottom to flow into a discrete layer of the lake as aninterflow. Some underflows may be dense enough to flow to the lowest point in a lake orto the dam that forms a reservoir. If the inflow is less dense than water at the lakes sur-

face, an overflow occurs. Fig. 5.3 illustrates these three types of inflows.At the plunge or separation point, the internal Fr of the stratified lake Frd is equal to

the Fr of the inflow at that point (Frp), as noted in Eq. Frd(5.38). If the difference in den-sity in between the lakes surface and the inflow is positive, an overflow occurs, andif is negative, an underflow occurs. If the slope of the reservoir bottom, or valley, ismild (SB 0.007), then the hydraulic depth (yo) is the normal depth of flow. For steepslopes (SB 0.007), the hydraulic depth is the critical depth (Akiyama and Stefan, 1984).

For tributary or river channels that are approximately rectangular or triangular, thehydraulic depth and location of the plunge point or separation point can be calculated. Fora rectangular cross section of constant width, the hydraulic depth is

yo 1/3

1/3

(5.41)

where Q the riverine inflow rate (m3/s) equal to VA, A the cross-sectional flow in areaof the river (m2), B the conveyance width (m), and q the flow per unit width (m2/s).Similar expressions were proposed by Akiyama and Stefan (1984), Jain (1981), Singh andShah (1971), and Wunderlich and Elder (1973), among and others. Savage and Brimberg(1973) developed an independent expression for the Froude number at the plunge point orpoint of separation (Frp) based on the conservation of energy and the theory of two-lay-ered flow in stratified water bodies, which can be expressed as

Frp

S

fb

b

0.478

(5.42)

where fb the dimensionless bed friction factor and fi dimensionless interfacial fric-tion. Martin and McCutcheon (1998) have illustrated the calculations and summarized the

validation of these equations by an example derived from Ford and Johnson (1981, 1983).For a triangular cross section with an angle 2 between the channel or valley walls, the

hydraulic depth is one-half the total depth. The area of the cross section (m2) is A y2

o

tan(), which, when substituted into the expression for the normal densimetric numberFrnand solved for the hydraulic depthyo (m), is

yo 0.5 1\5

where the bottom depth (distance between water surface and apex of the triangular crosssection) is twice the hydraulic depth for a triangular cross section. Hebbert et al . (1979)derived an expression for the downstream densimetric Froude number at the plunge pointor separation point Fp for normal flow (SB 0.007) in a triangular crosssection, relatedto the reservoir characteristics as

2Q2

Frn2g tan2 ()

2.05

1 f

f

b

i

q2

Frp2g

Q2

Fr2

p gB

2

5.16 Chapter Five





17/30

Fr2

n sin()

C

t

D

an(Sb) [1 0.85 C

1/2

D sin ()] (5.44)

where CD the dimensionless bottom drag coefficient [CD (fi fb)/4].Equations (5.42) and (5.44) are based on characteristics of the reservoir or lake. (SeeMartin and McCutcheon (1998) and Gu et al. (1996) for an example of the calcula-tions.)

5.5.3 Speed, Thickness, and Width of Overflows

Martin and McCutcheon (1998) have noted that the speed of an overflow (vofwith dimen-sions m/s) can be estimated from the celerity of a wave in a frictionless flow, but this con-sistently overestimates the rate of spread. Instead, Koh (1976) developed a more practi-

cal semiempirical expression based on uniform flow which reduces to (Ford andJohnson, 1983)

vof 1.04g yofwhere the thickness of the overflowyof (m) can be estimated from (Kao, 1976) as

yof 1.24

1/3 (5.46)

In natural settings, overflows are usually dissipated by mixing caused by wind and solarheating before traveling too far.

Horizontal spreading of an overflow is estimated using the inflow Fr defined by Eq.(5.38). Safaie (1979, cited in Ford and Johnson, 1983) found that for Frd 3, the flow isan unsteady, buoyancy-driven spread and can be assumed to be completely mixed lateral-ly except for abrupt changes in the entrance geometry. Typically, reservoirs widen gradu-ally where major tributaries enter, but lakes may have an abrupt widening at the mouth oftributaries. For Frd 3, the inflow acts like a jet that expands proportionally with distanceB(x) B0 cxwhere B(x) the overflow width (m) at distancexmeasured from the sep-aration point (m), B0 the width of the riverine or tributary flow at the separation point(m), and c a dimensionless empirical constant (Ford and Johnson, 1983). From labora-tory experiments with plane jets, the value ofc has been determined to be approximately0.16 (Fischer et al. 1979; Ford and Johnson, 1983).

5.5.4 Underflow or Density Current Mixing

Underflows are dominated by two mixing processes. First, significant mixing occurs dur-ing the plunge beneath the surface. Second, shear at the interface with ambient lake orreservoir water will result in mixing and entrainment as the underflow moves downward.

The initial turbulent mixing of the plunging flow will increase the total flow rate of theunderflow and reduce the density and concentration gradients. Thefraction entrainmentcaused by plunging is (Qp Q)/Q, where Qp is the flow rate at the plunge point (m

3/s) andQ is the river flowrate (m3/s). For mild slopes SB < 0.007, is on the order of 0.15(Akiyama and Stefan, 1984). The depth of the underflow is the normal depth of flow. Forsteep slopes SB > 0.007, is on the order of 1.18 and the density current depth is the crit-

q2

g






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5.18 Chapter Five

ical depth (Akiyama and Stefan, 1984). However, the entrained fraction is highly vari-able. The dilution of concentrations or temperatures resulting from mixing in plungingflows follows from a simple mass or heat balance

Cp C

1

a

C (5.47)

where Cis the inflow concentration (g/m3 or mg/L3) or temperature (C), Ca is the ambi-ent concentration (g/m3 or mg/L3) or temperature (C) of the lake, and Cp is the concen-tration (g/m3 or mg/L3) or temperature (C) of the plunging flow after initial mixing.

The mixing after plunging results from bottom shear as well as shear at the interfaceof the underflow with ambient lake water. For a triangular cross section, the entrainmentcoefficientis (Imberger and Patterson, 1981).

E 12 CkC

3/2D Fr

2b (5.48)

where laboratory experiments indicate that Ck is approximately 3.2 (Hebbert et al. 1979),CD the dimensionless bottom drag coefficient defined following Eq (5.42), Frb the inter-nal fronde number

Frb

u

b

bhb (5.49)

where ub underflow velocity, hb underflow depth, and b relative density differ-

ence. The entrainment coefficientEis a constant for a specific body of water.yuf (6/5)ExyO

The depth or thickness of the underflow (m) is a linear function of the entrainmentcoefficient (Hebbert et al. 1979; Imberger and, 1981), wherexis the distance downstreamfrom the plunge point (m) and yo is the initial thickness of the underflow (m) that isapproximately equal to the depth at the plunge point. If entrainment is limited, the depthof the underflow remains approximately constant as long as the bottom slope remains con-stant. The increase in flow rate because of entrainment for an underflow in a triangularcross section is solved iteratively as

Q(x) Q1

yy

u

1

5/3

1

(5.50)

where Q1 the discharge (m3/s) and y1 the depth (m) from the previous calculation

step. For the initial iteration, Q1 the discharge at the plunge point Qp (m3/s) andy1

the plunge point depthyo (m).Because of more significant differences in density and less internal mixing contrasted

with the epilimnion, underflows tend to remain more coherent than overflows. Sedimentladen underflows, especially, tend to travel to the lake outlet or dam.

5.5.5 Interflow Mixing

After experiencing approximately 15 percent entrainment at the plunge point (for mildslopes) and mixing as an underflow, an interflow intrudes into a lake at the depth at whichneutral buoyancy is achieved. The turbulence generated by bottom shear is dissipatedquickly, and entrainment into the interflow is dominated by interfacial shear with ambientlake water above and below the intrusion layer.





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When the momentum of inflow is small, an interflow is analogous to a withdrawalfrom a dam discussed in Sec. (5.5.6). Interflows are governed chiefly by three conditionsbased on the dimensionless numberR FriGr

1/3 where Fri is the internal Fr defined in Eq.

(5.51) and Gr is the Grashof number (Gr), both of which are computed at the depth ofintrusion. The internal Froude Number computed at the intrusion depth is

Fri N

q

LI

2

I

BI

Q

N

I

L2

I

(5.51)

where qI the interflow rate per unit width following entrainment at the intrusion point( m2/s), LI the length of the reservoir at the level of intrusion (m), QI the interflow rate(m2/s), BI the intrusion width (m), andN the buoyancy frequency (s

-1) expressed as

N

gIyI

I (5.52)where I density difference between the layers into which the flow is intruding(kg/m3), I density of the intrusion (kg/m3), andyI the thickness of the depth of theintrusion (m). The dimensionless Grashof number Gr is the square of the ratio of the dis-sipation time to the internal wave period or

GrN

2

2

L

v

4

I (5.53)

where v the vertically averaged diffusivity (m2/s). Generally, ifGr 1, then an inter-

nal wave field will decay slowly, but ifG

r 1 then viscous dissipation damps wavesquickly (Fischer et al. 1979). Imberger and Patterson (1981) also introduced a dimen-sionless time variable

t* G

t

r

N1/6

where t time(s), which, along with the Prandtl number Pr v/t, where t is vertical-ly averaged diffusivity of heat (m2/s), is used to define three interflow conditions:

1. If R 1, the intrusion is governed by a balance of the inertial and buoyancy forcesso that the actual intrusion length Li is proportional to time, as given by (Ford andJohnson, 1983; Imberger et al., 1976).

Li 0.44Li Rt* 0.44 qINt (5.54)

If the speed of the intrusion is constant or uniform, the velocity vI is Li/t, so that

vI 0.44 qIN 0.194

gI

IyI

1/2

where m the density of the intrusion. The difference in density in the computationof the buoyancy frequency is that occurring over the thickness of the intrusion hm,

which, along with the relationship um qm/hm, can be substituted into the above equa-tion to yield an alternative formulation for the speed of intrusion. The thickness of theinterflow can be solved by assuming uniform flow (Ford and Johnson, 1983).

hm 2.99

1/3

(5.55)q

2

m

gm

m





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5.20 Chapter Five

where hm is generally distr ibuted equal ly above and below the center line of theintrusion.

2. If R t*

R P

2/3

r , then the flow regime is dominated by the balance between vis-cous and buoyancy forces and the intrusion length becomes

Li 0.57 L R2/3

t*5/6

(5.56)

The thickness of the interflow is

hm 5.5LmGr1/6 (5.57)

In this regime, the flow is generally distributed so that 64 percent lies above the cen-ter line of the intrusion (Imberger, 1980); thus, the half-thickness (hma) of the interflow

above the center line is given by

hma 3.5LmGr1/6 (5.58)

and the half-thickness below the center line is given by

hmb 2.0Lm Gr1/6 (5.59)

3. If P2/3

r t* R

-1then the flow regime is dominated by viscosity and diffusion and

the intrusion length becomes

Li= C

LR3/4

t*3/4

(5.60)where C

is a coefficient, that generally is unknown (Fischer et al. 1979).

Ford and Johnson (1986) indicated that unless dissolved solids dominate the densi-ty profile (i.e., Pr is high), intrusions into most reservoirs haveR 1, where inertia andbuoyancy dominate. Because the difference in density varies with the location of thelimits of the interflow zone above and below its center line, the solution proceeds byestimating a value ofhm and then by computing the difference in density, which is thenused to compute a revised estimate ofhm. This process is repeated until convergenceoccurs.

The equations for intrusion require information on both the morphometry of the reser-voir and the temperature distribution. The widths used in the formulations should repre-sent the conveyance width (Ford and Johnson, 1983). Because the time for the intrusionto pass through a lake can be relatively long, the flow rates used in the calculationsshould represent an average value over the period of intrusion. To estimate the time scalein their analysis of intrusions in DeGray Lake in Arkansas, Ford and Johnson (1983) usedthe length of the lake and m and hm across the thermocline. For DeGray Lake, the intru-sion time scale ranged from 4 to 6 days. Changes in outflow during the period of theintrusion also can affect the movement through the lake. Interflows may stall and col-lapse if the inflow or outflow ends. Interflows also may be diverted or mixed because ofchanges in meteorological conditions that influence epilimnion mixing and thermoclinedepth.

The temperature or density of the interflow will remain constant. However, the inter-flow will spread laterally and the thickness will increase caused by entrainment of ambi-ent water. The resulting concentrations can be computed from a mass balance vinCBhn constant, where vin is the velocity of the interflow; C the concentration or temperature;B the reservoir width, which may vary with distance from the separation or detachmentpoint; and hn the thickness of the interflow.





21/30


FIGURE5

.4

Reservoirwithdrawal.(Adapted

fromMartinandMcCutcheon,19

98)





22/30

5.22 Chapter Five

5.5.6 Outflow Mixing

The withdrawal velocity profile is used in models CE-QUAL-R1 (Environmental

Laboratory, 1985) and CE-QUAL-W2 (Cole and Buchak, 1993) and in calculations to pre-dict the effects of withdrawals on reservoir and tail race water quality. The extent of awithdrawal zone (Fig. 5.4) strongly depends on the ambient lake stratification and releaserate, location of the withdrawal, and reservoir bathymetry. For a given outflow rate andlocation, the withdrawal zone thins as the density gradient increases. Depending on thedegree of stratification, withdrawal rate and location, and other factors related to thedesign of the dam and the bathymetry of the reservoir, the withdrawal zone may be thinor may extend to the reservoir bottom or water surface. Within the withdrawal zone, thevelocity distribution will vary from a maximum velocity to zero at the limits of the zone,depending on the shape of the density profile. The maximum velocity is not necessarily

centered on the withdrawal port.A number of methods predict the extent of withdrawal zones and the resulting veloci-ty distributions. Fischer et al. (1979) described methods of computing withdrawal patternssimilar to those used in the analysis of interflows in the previous section. The BoxExchange Transport, Temperature, and Ecology of Reservois (BETTER) model and theSELECT model based on the original work of Bohan and Grace (1973) are the more prac-tical approaches. The BETTER model, applied to a number of Tennessee Valley Authorityreservoirs, computes the thickness of the withdrawal zone above and below the outlet ele-vation from y cw Qout, where Qout the total outflow rate and cw is a thickness coeffi-cient. The model assumes a triangular or Gaussian flow distribution to distribute flowswithin the withdrawal zone (Bender et al. 1990).

The SELECT model (Davis et al. 1985) computes the in-pool vertical distribution ofoutflow and concentrations of water quality constituents, the outlet configuration anddepth, and the discharge rate (Stefan et al. 1989). The SELECT code also is applied assubroutines in generalized reservoir models, such as CE-QUAL-R1 (EnvironmentalLaboratory, 1985). The model is based on the following equations.

The theoretical limits of withdrawal (Bohan and Grace, 1973) were modified by Smithet al. (1985) to include the withdrawal angle as

Z

Q3

o

N

ut

(5.61)

where Z distance from the port center line to the upper or lower withdrawal limit; the withdrawal angle (radians); andN the buoyancy frequency [g/(Z)]1/2, in which the difference in density between that at the upper or lower withdrawal limit and atthe port centerline; and the density (kg/m3) at the port center line. The convention isthat is positive for stably stratified flows such that (upper limit) (with-drawal port) or (withdrawal port) (lower limit). The elevation of the watersurface, the bottom, of the reservoir, and the withdrawal port and the density profile mustbe known. The equation must be solved iteratively since both the distance from the portcenter lineZand the density as a function ofZare unknown. A typical solution procedurewhere the upper and lower withdrawal zones can form freely within the reservoir withoutinterference at the surface or bottom is as follows:

1. Rearrange the equation as QoutZ3N/ 0.

2. Check to see if interference exists by, first, usingZequal to the distance from theport,s center line to the surface. Estimate the density at the center line of the with-drawal port and the water surface and substitute the values into the rearrangedequation. If the solution is not-zero and is positive, surface interference exists.





23/30


FIGURE

5.5

Definitionofwithdrawalcharacteristics.(FromMartinandMc

Cutcheon,1998)





24/30

Similarly, substitute the distance from the port center line to the bottom, along withthe density at the bottom of the reservoir, and determine if a bottom interferenceexists.

3. If both of the evaluations from Step 2 are negative, the withdrawal zone formsfreely in the reservoir. The limit of the surface withdrawal zone above the portcan be determined by using iterative estimates of values forZand the density atthe height above the center line until the equation approaches zero to within sometolerance. The lower limit of withdrawal below the port center line can be deter-mined in a similar manner.

4. If surface or bottom interference exists, a theoretical withdrawal limit can bedetermined using values ofZcomputed using elevations above the waters sur-face for surface interference or below the reservoirs bottom for bottom interfer-

ence. However, this solution requires an estimate of density for regions outsidethe limits of the reservoir. Davis et al. (1985) estimated these densities by linearinterpolation using the density at the port center line and the density at the sur-face or bottom of the reservoir.

For the case where one withdrawal limit intersects a boundary and the other does not,the freely forming withdrawal limit cannot be estimated precisely using the rearrangedequation. Smith et al. (1985) proposed an extension to estimate the limit of the freelyforming layer similar to that described above

Q

N

out

0.125(D

d)3

1 1 sin

D

D

d

D

D

d

, (5.62)

where d the distance from the port center line to the boundary of interference (m) andD the distance between the free withdrawal limit and the boundary of interference (m)shown in Fig. 5.5. The length scale in the buoyancy frequency Nis D in place ofZ, and is the difference in the density between that at the surface for withdrawals that extendto the surface and between the lower free limit or density at the bottom for withdrawalsthat extend to the bottom and upper free limit. For consistency with the definition of sta-ble stratification as positive, the convention is that (surface layer) (free limit)or (upper free limit) (bottom layer).

Once the limits of withdrawal are established, the distribution of withdrawal veloci-ty is estimated by dividing the reservoir into layers, the density of which is determinedat the center line of each layer. The computation of the vertical velocity distribution isbased on the location of the maximum velocity, which can be estimated from (Bohanand Grace, 1973).

YL Hsin1.57

Z

H

L

2

(5.63)

where YL the distance from the lower limit to the elevation of maximum velocity (m)

shown in Fig. 5.4, H the vertical distance between the upper and lower withdrawal lim-its (m), andZL the vertical distance between the outlet center line and the lower with-drawal limit (m). If the withdrawal intersects a physical boundary, the theoretical with-drawal limit is used, which may be above the waters surface or below the reservoirs bot-tom. Once the location of the maximum velocity Vmax (m/s) is determined, the normalizedvelocity VN(I) V(I)/Vmax in each layerIis estimated for withdrawal zones that intersecta boundary as (Bohan and Grace, 1973).

5.24 Chapter Five





25/30

VN(I) 1

y

Y

(

L

I)

m

(

a

I

x

)

2

(5.64)

or for a withdrawal that does not intersect a boundary

VN(I) 1

2

(5.65)

where V(I) the velocity in layerI(m/s), y(I) the vertical distance from the elevationof maximum velocity to the center line of layerI(m), YL the vertical distance from theelevation of maximum velocity to the upper or lower withdrawal limit (m) determined bywhether the centerline of layerIis above or below the point of maximum velocity, (I) the density difference between the elevation of maximum velocity and the center lineof layerI, and max the difference in density between the point of maximum velocityand the upper or lower withdrawal limit.

If the withdrawal intersects the surface or the bottom, velocities are calculated for loca-tions either above the waters surface or below the reservoirs bottom and the distributionis truncated at the reservoirs boundaries to produce the final velocity distribution. Theflow rate in each layerIis

q(I) Qout (5.66)

where Qout

the total release rate and m the number of layers. The quality of the releasecan be determined from a simple flow-weighted average or mass balance as

CRV (5.67)

where CR the concentration or temperature of waterquality constituent C in therelease and C(I) the concentration or temperature in each layer.

For discharge over a weir, the withdrawal limit Zand average velocity in the with-drawal zone Vweir is derived from the densimetric Froude number [Eq. 5.38] as (Grace,1971, Martin and McCutcheon, 1998),

0 Vweir C1g

H

(Z

w

Hw)2

C2(Z

Hw) (5.68)

where the difference in density between the weir crest and the lower withdrawallimit, the density at the weir crest elevation, Hw head above the weir crest eleva-tion, Z distance between the crest elevation and the lower withdrawal limit, and C1 andC2 are constants, which have values of

C1 0.54 and C2 0 forZ

Hw

Hw

2.0

and (5.69)

C1 0.78 and C2 0.70 forZ

Hw

Hw 2.0

q(I) C(I)

NI 1

q(I)

VN(I)mI= 1

VN(I)

y(I) (I)

YL MAX






26/30

5.26 Chapter Five

5.5.7 Mixing Caused by Meteorological Forces

Windgenerated waves and convective cooling cause significant mixing at the water sur-

face. Wind shear causes waves at the surface and at each density interface within a lake orreservoir, such as the thermocline, and larger scale surface mixing by Langmuir circula-tion results from sustained wind. Wind setup, seiche, and upwelling are caused by mete-orological events that generate mixing over much larger areas. Internal waves are causedby shearing currents set up by both wind and other currents and, although not as obviousas surface waves, these can be larger and more effective in causing mixing. The intensityof wave mixing and turbulence is a direct result of wind energy or the energy in othershearing currents.

The basic characteristics ofwaves are amplitude or height between trough and crestand the length between crests. The wave period is the time required for successive waves

to pass a given point. Progressive waves move with respect to a fixed point, whereas stand-ing waves remain stationary while water and air currents move past. The height and peri-od of wind waves are related to wind speed, duration, and fetch. Fetch is the distance overwhich the wind blows or causes shear over the waters surface. As fetch increases, thewavelength increases; long wavelengths are only produced in the presence of a long fetch.The shortest wavelengths require only limited contact between wind and water. Waveswith a wavelength less than 2 cm (6.28 cm) are capillary waves, which are not importantin the modeling of lakes and reservoirs. The more important gravity waves have wave-lengths longer than 2 cm. The two types of gravity waves are short waves and longwaves, distinguished by the interaction with the benthic boundary. The wavelength ofshort waves seen by eye on lakes and reservoirs is much less than the waters depth, and

they are not affected by bottom shear. Long waves, such as lake seiche, are influenced bybottom friction. Seiches are periodic oscillations of the waters surface and density inter-faces resulting from a displacement.

Shortwave motion is circular in a vertical plane, making a complete revolution as eachsuccessive wave passes. The orbital motion mixes surface layers or layers at an interface.With no net advection of water, the overall effect is dispersive. Thus, the mixing terms intransport and water quality models are generally increased to account for wave mixing,especially in the epilimnion. In a few cases, specific mixinglength formulas Kent andPritchard, 1957, Rossby and Montgomery, 1935; were derived for wave mixing, but theseformulas have not been applied in current models of water quality. No appreciable orbital

motion occurs below a depth of approximately one-half the wavelength in unstratifiedflow, a depth referred to as the wind mixed depth. The wind mixeddepth increases withfetch because the wave height and wavelength increase with increasing fetch. This is illus-trated by a simple relationship discovered by Lerman (1978) relating fetch to the depth ofthe summer thermocline for a wide variety of lakes of different sizes and shapes.

As wavelength becomes longer in relation to the depth, or as water becomes shallow-er, wave orbits become increasingly flatter or elliptical. As the orbits flatten, the motionof the water essentially becomes horizontal oscillation (Smith, 1975) so that the motionof the water caused by, long waves is more advective rather than dispersive. For longwaves, the wave speed or celerity is c (gY)0.5.

As short waves enter shallow water, the bottom affects orbital motion. From this pointinland to the line where wave breaking occurs, the depth is less than onehalf the waveperiod. In this shore zone, wave velocity decreases with the square root of the depth,which results in a corresponding increase in wave height. Waves distort as water at thecrest moves faster than the wave, creating an instability. These unstable waves may even-tually collapse, forming breakers or whitecaps, depending on the wave steepness of thewaves, the wind speed and direction, the direction of the waves, and the shape and rough-





27/30


ness of the bottom. A spilling breakertends to form over a gradually shoaling bottom andtends to break over long distances, with the wave collapsing downward in front of thewave. Plunging breakers occur when the bottom shoals rapidly or when the direction of

the wind opposes the wave. The plunging breaker begins to curl and then collapses beforethe curl is complete. A plunging or surging breaker does actually not break or collapse butforms a steep peak as the wave moves up the beach. The type of breaking wave and theassociated energy controls beach erosion, aquatic plant growth, surf-zone mixing, and theexchange of contaminants between surface and ground waters.

After breaking, waves continue to move up a gradually sloping beach until the force ofgravity forces the water back. The extent to which the water runs up the beach is calledthe swash zone. The movement of the swash up the beach may result in the deposition ofparticles and debris, causing swash marks at the highest point of the zone. Wave runupin the swash zone also sets up an imbalance of momentum along the porous beach face

that pumps contaminants into and out of the beach (McCutcheon, 1989).In large lakes and reservoirs with an extremely long fetch, parallel pairs of large verti-

cal vortices or circulatory cells known asLangmuir circulation develop at an angle of 15clockwise with the general direction of a sustained wind, when wave and current condi-tions are favorable. The depth of the vortices depends on stratification and may interactwith internal waves formed on the thermocline, deepening over the troughs of internalwaves. Where the counterrotating Langmuir cells converge, visible streaks or bands formon the surface that tend to accumulate floating debris. In the convergence zone, downwardvelocities of 26 cm/s carry surface waters toward the thermocline. These downward cur-rents move in a circular fashion and turn upward into a divergence zone midway betweenthe Langmuirstreaks. Water near the thermocline moves to a zone near the surface at avelocity of about 1 to 2 cm/s over a larger area. As first proposed by Langmuir (1938), thistype of large-scale circulation also contributes to the vertical mixing of the epilimnion.Like smaller-scale orbital wave mixing, the effect of Langmuir circulation is lumped intovalues selected for the eddy viscosities and eddy diffusivities of the epilimnion.

Because of the smaller differences in density across density interfaces within a body ofwater, internal waves travel more slowly than do surface waves, but they achieve greaterwave heights.Internal waves include standing waves, such as seiches (Mortimer, 1974) andinternal hydraulic jumps (French, 1985), but most are progressive waves that radiate energyfrom the point at which the waves were generated (Ford and Johnson, 1986). Wind shear,water withdrawals, hydropower releases, and thermal discharges as well as local distur-

bances produce internal waves. The most significant mixing between stratified layers occurswhen internal waves break (Turner, 1973). Before breaking, internal waves mix the wateradjacent to the interface and sharpen the density interface to increase the likelihood of break-ing. When wave breaking does occur, the entrained water is mixed through the adjacent layer.

Among the most important internal waves is the seiche. As defined above, seiches areperiodic oscillations of the water surface and density interfaces resulting from a displace-ment. Displacements are typically caused by large scale wind events or large withdrawals.Sustained wind across a lake surface increases the elevation the waters surface at thedownwind boundary of the lake, causing wind setup. As the wind subsides, the waterssurface tilt or displacement results in a sloshing motion, or seiche, of the lake surface andin thermocline if the lake is stratified. If hydropower operations or reservoir releaseschange the net flow toward the dam, the water piles up at the dam and forms a seiche,often resulting in noticeable differences in thermocline depths between periods of opera-tion and nonoperation, such as between weekdays and weekends. More rarely, a seichemay result from earthquakes or other geologic events. During the rocking or sloshing,potential energy is converted to kinetic energy and is dissipated by bottom friction.





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5.28 Chapter Five

Wind setup in Lake Erie may exceed 2 m during severe storms (Wetzel, 1975), but fora moderate storm blowing over the long axis of Green Bay, Wisconsin the wind setup hasreached approximately 12 cm (Martin and McCutcheon, 1998). An estimate of wind setup

can be obtained from the onedimensional equation of motion assuming constant depth,negligible bottom stress, and steadystate conditions in an unstratified lake, or

x

ag

C

D

y

u2

w

g

v

y

2

* (5.70)

where the deviation of the waters surface (m), x the horizontal distance (m), a the density of air (kg/m3), the density of water (kg/m3), CD the dimensionless dragcoefficient, uw the wind speed (m/s), y the water depth (m), and v* the frictionvelocity in water (m/s) or (s/)0.5, in which s is the surface shear stress (kg/m s2).The term /xis positive in the direction of the wind. The divergence between wind andshear force is negligible in shallow lakes and reservoirs but not in deep oceans.

5.6 PLUME AND JET HYDRAULICS

A jetis the discharge of a fluid from an opening into a large body of the same or similar fluidthat is driven by momentum. Aplume is a flow that, while resembling a jet, is the result ofan energy source providing the fluid with positive or negative buoyancy rather than momen-tum relative to its surroundings. Many discharges into the environment are discussed interms of negatively or positively buoyant jets, implying that they derive from sources thatprovide both momentum and buoyancy. In such cases, the initial flow is driven primarily bythe momentum of the fluid exiting the opening; however, if the exiting fluid is less or moredense than the surrounding fluid, it is subsequently acted on by buoyancy forces.

Jets and plumes can be classified as either laminar or turbulent, with the differencebetween the two being described by a Reynolds number, as with pipe flow. Near thesource of the flow, the flow of a jet or plume is controlled entirely by the primary initialconditions that include the mean velocity of the jets exit, the geometry of the exit, and theinitial difference in density between the discharge and the surrounding, or ambient, fluid.Secondary initial conditions include the intensity of the exiting turbulence and the distri-bution of the velocity. Following Fischer et al. (1979), the factors of prime importance tojet dynamics can be defined as follows:

1. Mass flux the mass of fluid passing a jet cross section per unit time

mass flux = aA

(u)dA (5.71)

where A is the cross-sectional area of the jet and u the time-averaged velocity of thejet in the axial direction.

2. Momentum flux amount of momentum passing a jet cross section per unit time

momentum flux =

a

A

(u2)dA (5.72)

3. Buoyancy flux buoyant or submerged weight of the fluid passing a jet cross sec-tion per unit of time

buoyancy flux = aA

(gu)dA (5.73)

where the difference in density between the surrounding fluid and the fluid in thejet. It is convenient to define g()/g g' as the effective gravitational acceleration.





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5.6.1 Simple Jets

The two dimensional or plane jetissuing from a slot and the round jet issuing from a noz-

zle into a quiescent ambient fluid are among the simplest cases of jets that can be consid-ered. These jets have been studied extensively, and there is a reasonable understanding of

how they behave. The boundary between the ambient and jet fluids is sharp at any instant,

and if a tracer were present in the jet fluid, time-averaged measurements would show a

Gaussian distribution of tracer concentration (C) across the jet or

CC

m

expkj

yx

2

(5.74)

where the subscript m the value ofCon the jet axis, x the distance along the jet axis,

kj experimental coefficients, and y the transverse (or radial) distance from the jetaxis. The Gaussian distribution also is valid for the time-averaged velocity profile across

the jet provided that the measurement is taken downstream of the zone of established flow.

In the case of a circular jet, the length of thezone of established flow is approximately 10

orifice diameters downstream.

Downstream of the zone of established flow, the jet continues to expand and the mean

velocity and tracer concentrations decrease. Within the zone of established flow, the

velocity and concentration profiles are self-similar and can be described in terms of a

maximum value (measured at the jets center line) and a measure of the width or, in the

case of the velocity, distribution:

vv

m

f

b

y

w

(5.75)

where vm the value ofv on the jets center line, y a coordinate transverse to the jets

axis, and bw the value ofxat which v is some specified fraction ofvm (often taken as

either 0.5 or 0.37; Fischer et al. 1979). The functional form offin Eq, (5.75) is most often

taken as Gaussian.

Almost all the properties of turbulent jets that are important to engineers can be

deduced from dimensional analysis combined with empirical data (Fischer et al. 1979).

These results are summarized in Table 5.2.

5.6.2 Simple Plumes

Because the simple plume has no initial volume or momentum flux (e.g., smoke rising

from a fire), all variables must be a function of only the buoyancy flux (B), the vertical

distance from the origin (y), and the viscosity of the fluid where

B g

o

Q g'

oQ (5.76)

and o difference in density between the plume fluid and the ambient fluid and go apparent gravitational acceleration.

Results similar to those for jets are summarized in Table 5.3, and the numerical con-stants given are from Chen and Rodi (1976).





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5.30 Chapter Five

TABLE 5.3 Summary of Plume Properties

Parameter Round Plume Plane Plume

Maximum time-averaged (4.7 0.2)B1/3y-1/3 1.66 B1/3

velocity

vm

Maximum time-average (9.1 0.5)M B-1/3y-5/3 2.38M B-1/3B-1

tracer concentration

Cm

Volume flux (0.15 0.015)B1/3y5/3 0.34 B1/3y

Q

Ratio 1.4 0.2 0.81 0.1

Cm/Cavg

Source: After Fischer et al. 1979.

TABLE 5.2 Summary of the Properties of Turbulent Jets

Parameter Round Jet Plane Jet

Initial volume flow rate D

4

2V0 b0y0 V0

Qo

Initial momentum flux D

4

2V

2

0 b0y0V

2

0

Mo

Characteristic length scale M

Q

0

0

M

Q2

0

0

lQ

Maximum time-averaged

velocity vm MQ (7.0 0.1)

ly

Q

vm M

Q (2.41 0.04)

ly

Q

Vm

Maximum time-averaged

tracer concentration C

Cm

0

(5.6 0.1)

l

yQ

C

Cm

0

(2.38 0.04)

l

yQ

Cm

Mean dilution Q

Q

0

(0.25 0.01)

l

y

Q

Q\Q0 (0.50 0.02)

l

y

Q

Ratio

Cm

/Cav

1.4 0.1 1.2 0.1

Source: After Fischer et al. 1979.


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Chapter 5 – Environmental Hydraulics