TWO-DIMENSIONAL NUMERICAL 0OCEAN THERMAL POWER PLANT.

TWO-DIMENSIONAL NUMERICAL MODEL

OF THE NEAR-FIELD FLOW FOR AN0OCEAN THERMAL POWER PLANT.

PART II. SIMULATION OF THE

LOCKH EED BASELINE DESIGN.

SAI-76-625-WA C

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NRL-GFD/OTSC 6-76

TWO-DIMENSIONAL NUMERICAL MODEL

OF THE NEAR-FIELD FLOW FOR AN OCEAN THERMAL POWER PLANT.

PART 1I. SIMULATION OF THE LOCKHEED BASELINE DESIGN.

Glyn 0. Roberts

Fluid Mechanics Division,

Science Applications, Inc.

and -, .

Steve A. Piacsek* and Juri Toomre**

Geophysical Simulation Section, Code 7750

Naval Research Laboratory

SAI-76-625-WA

*fov at Naval Ocean Research and Development Activity

**Department of Astrogeophybics, University of Colorado

Supported by the Ocean Thermal Energy Conversion Program,

Division of Solar Energy, Energy Research and DevelopmentAdminitration, under ERDA contract 1 (49-26) 1003.

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20. ABSTRACT (continued)

50 ft, 100 ft, 200 ft, and 1400 ft are separately assumed; ve bei.evw-th theS100-ft and 200-ft computations essentially bracket the results for the proto-

type flow.

lwlz-imerical rpylts fr the !Wjage warm Inflow temperatures, in the -tsrwE)cases, are-75.7 F,Jj.-2T 77.9 F, and 78.6"7. This is wiM a surfacetemperature of 80 F.° The warm o ty- is defined to be 3Fooler; the coldoutflow temperature is set to 155 F.--

ThEA'.umerical results for the OTP-generated turbulent diffusivity indicatethat it is effectively confined to the neighborhood of the plant. Its valueis very small outside such a neighborhood. I.-Conclusions are drawn with regard to OTPP operation and modeling.

I

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TABLE OF CONTENTS

ABSTRACT ........................ . . . . . . . . . ±l

1. INTRODUCTION ......... ....................... -iC.

2. NRFL02 MODEL OF THE LOCKHEED NEAR-FIELD FLOW CONDITIONS • 2-1

2.1 The Lockheed Baseline OTPP Design

2.2 IRFL02 Model of the Lockheed Design

2.3 MUFL02 Turbulence Model

2.4 The Assumed Ambient Ocean Conditions

3. RESULTS AD DISCUSSION .................... 3-1

3.1 Numerical Results

3.2 Validity of the Results

3.3 Conclusions

ACxNOWLEDGE4ENTS .. .. .. .. .. .. .. . . .. ... A-1

REF ERECES . . . . . . . . .. .. .... R-1

i

0

ABSTRACT

This report describes a computation of the stratified

turbulent flow-field near the Lockheed baseline Ocean Thermal Power

Plant (OTPP) design, with zero ambient current. The calculation uses

our two-dimensional near-field computer code NRFL02, and approximates

the flow associated with one of the four power modules by assuming a

rectangular domain of depth 500 ft, with horizontal slots at one end

to represent the warm inflow and the two "utf-iows. We assume that

* - the flow is statistically uniform across the width of this domain.

Widths of 50 ft, 100 ft, 200 ft, and 400 ft are separately assumed;

we believe that the 100 ft and 200 ft computations essentially

bracket the results for the prototype flow.

The numerical results for the average warm inflow

temperatures, in the four cases, are 75.TF, 77.2 F, 77.9°F, and

78.60F. This in with a surface temperature of 800F. The warm00

outflow is defined to be 3°F cooler; the cold outflow temperature

is set to 45 F°

The numerical results for the OTPP-genersted turbulent

diffusivity indicate that it is effectively confined to the neighbor-

hood of the plant. Its value is very small outside such a neighbor-

hood.

Conclusions are drawn with regard to OTPP operation and

modeling.

iii

1. INTRODUCTION

We have discussed the external flow problems associated

with ocean thermal power plant (OTPP) operation in previous reports

(Piacsek, Toomre, and Roberts, 1975, and Piacsek, Martin, Toomre, and

A Roberts, 1976). The first problem concerns the warm and cold OTPPinflows. With a poor design, there is strong turbulent mixing and

recirculation of outflow water, resulting in a substantial reduction

in the temperature difference constituting the thermal resource.

The second problem concerns the environmental impact of OTPP operation;

large-scale implementation in limited ocean regions may produce

significant thermal and other changes, either beneficial or otherwise.

Our computer code NRFL02 for calculating the near-field

turbulent flow and the associated temperature distribution is described

in Parts I and III of this report series. The first objective of

these external flow computations is the determination of the OTPP

inflow temperatures, as modified by turbulent mixing and recirculation

of outflow water. The second objective is to obtain results which

can be used in determining the far-field environmental impact of OTPP

operation.

NRFL02 considers a rectangular two-dimensional domain,

with outflows and inflows on the left vertical boundary to model the

OTPP intakes and outflows. The surface and bottom boundaries are

impermeable, while the passive boundary conditions at the right allow

horizontal inflow from and outflow to the ambient ocean. A first-

order turbulence closure model is used, with a turbulent kinetic

energy equation having generation, decay, advection, and diffusion

terms. This kinetic energy is used with an imposed length-scale and

the local density stratification to calculate the single turbulent

diffusivity coefficient, which is used in the equations for the

horizontal motion, the vertical motion, the temperature, and the

turbulent kinetic energy. Where the horizontal flow is into the

computational domain, either on the left (OTPP) boundary or on the

1-1

right boundary with the ambient ocean, boundary values of all variables

are required. When the flow is out of the computational domain,

passive boundary conditions on the normal derivatives are required

for the vertical motion, the temperature, and the turbulent kinetic

energy.

In this report, we describe the application of NRFL02 to

the baseline OTPP design described by Lockheed (Trimble, 1975). This

design is probably more amenable to two-dimensional external flow

computation than the others described to date. In a sense, this can

be regarded as our baseline near-field study; in Part I, we applied

NRFL02 to a proposed two-dimensional laboratory simulation of

stratified turbulence driven by inflows and outflows.

NRFL02 is the first in a proposed series of near-field

computer codes, with increasingly complicated and flexible geometries,

OTPP specifications, and numerical methods. We hope to finalize the

turbulence closure model in the next code, NRFLOA (with an axisymetric

geometry). The coefficients will be tuned to give agreement with

experimental results obtained by other ERDA contractors. Three-

dimensional flow fields will then be studied using the code NRFL03.

In Section 2, we describe our model of the Lockheed

baseline design and of the ambient ocean, within the constraints of

IRFLO2. The spar design has an axisymmetric cold water pipe andV intake structure, with four power modules symmetrically placed around

it. We model the flow for a single module (in a 90 quadrant, with

the module at the center) in our two-dimensional code by assuming

a three-dimensional rectangular region, with uniform conditions across

a fixed width w to represent the azimuthal coordinate. Our leftL boundary is at a radius of 233 ft from the OTPP axis, and we separately

compute w values of 50, 100, 200, and 400 ft. The values 50 and 400

are extreme; we believe that the closest approximation is obtained

using a w value in the intermediate range. We model only the upperC 500 ft of the ocean, so that the deep cold inflow is excluded from

these computations.

C 1-2

Our numerical results are presented and discussed in

Section 3, and conclusions are drawn with regard to the Lockheed

design. The results are presented using computer graphics and

comparison tables. Even for the smallest w value considered, the

loss in warm inflow temperature through turbulent mixing and outflow

recirculation is only about 3°F. This loss would be significant, but

not disastrous to OTPP operation.

1-3

2. NRFL02 MODEL OF THE LOCKHEED NEAR-FIELD FLOW

CONDITIONS.

2.1 THE LOCKHEED BASELINE OTPP DESIGN.

The Lockheed design is described, in a slightly simplified

form, in Figure 2.1 and in the associated caption. In the spar

design, a cylindrical body 445 ft high, with a diameter of 200 ft,

extends downward from 100 ft below the surface. The cold water pipe

is a vertical cylinder 1000 ft long, with outside diameter decreasing

from 129 ft to 105 ft, attached at the bottom. The superstructure

is a vertical cylinder 180 ft long, with diameter 60 ft, attached to

the top, and extending 80 ft above the surface. The warm inflow is0a 45 cone, with bottom diameter 170 ft and height 55 ft, extending

up from the cylindrical OTPP body and meeting the superstructure

cylinder at a depth of 45 ft.

Four power modules are symmetrically placed around the

central spar. Each module consists of an evaporator cylinder, of

diameter 72 ft and length 133 ft, with a horizontal axis at depth

150 ft, and a similar condenser cylinder, at depth 315 ft, canted

downward at 100 at the outflow end. Joining these cylinders is a

vertical cylinder containing the turbines and generators and

auxiliary equipment. Crew access is provided through a thin

horizontal cylinder connection to the plant body. The warm

evaporator flow of 12000 cu ft/sec enters from the plant body and

flows out horizontally after being cooled by 3°F (Lockheed state a

slightly lower figure). The cold condenser flow of 15000 ft3/sec is

* warmed by about 2.40F and flows out with a direction 100 below the

vertical. The two outflows, for each model, are at a horizontal

distance of 233 ft from the OTPP axis, and at depths of 150 ft and

315 ft

2-1

FIGURE 2.1. The Lockheed Baseline OTPP Design

This is a side view, with all dimensions in feet; a top

view is shown in Figure 2.2. Four power modules are symmetrically

placed around the plant body, a vertical cylinder. The warm inflow

of 48000 cu ft/sec enters through the conical superstructure aroundthe plant body, is slightly cooled in the four evaporators, and

exists horizontally at depth 150 ft. The cold inflow of 60000 cu ft/sec

enters the cold water pipe at a depth of 1545 ft, is slightly warmed

in the four condensers, and exits at a depth of 315 ft, directed 100

below the vertical. The evaporators and condensers and the outflow

ports all have diameter 72 ft.

2-

2-2 ;

C. 233

10085

30warm Power

Axis Inflow Module

30233

85 nl

233

FIGURE 2.2. Top View of the Lockheed Baseline OTPP Design.

A side view is shown in Figure 2.1. All dimensions are in feet. The

four power models are attached to a central cylinder of diameter

200 ft. The warm inflow filter in a segment of a 43 0 cone, with

bottom diameter 170 ft, meeting the superstructure cylinder, with

diaseter 60 ft, at a depth of 45 ft.

2-4C

The above description is only slightly simplified. We

have omitted buoyancy tanks on the side of the main plant body,

between the power modules. Similarly we have omitted from the

description the buoyancy tanks on the top and bottom of each power

module. Otherwise the description is essentially complete.

2.2 NRFL02 MODEL OF THE LOCKHEED DESIGN.

As described in Part I, NRFL02 assumes a rectangular

domain, with statistical uniformity across a constant width in the

horizontal direction "into the paper". Thus the calculation is two-

dimensional. The normal flow distribution is imposed on the left

boundary, and is either zero, or into the region (representing an

OTPP outflow), or out of the region (representing an OTPP inflow).

For outflows, the temperature, vertical motion, and turbulence

intensity are also imposed, to complete the modeling of any

particular OTPP design.

Thus, to model the Lockheed design, we must first assume

a width over which the flow is statistically averaged. The mean flow

variables are assumed to be uniform over this width. We have

confined attention to the external flow for a single module of the

Lockheed design, and to its surface inflow and two outflows. Thus,

the inflow flux is 12000 cfs, and the outflow fluxes are 12000 cfs

and 15000 cfs. We have modeled these inflows and outflows as

occurring through horizontal slots on the left boundary of our

computational domain, centered at depths 75 ft, 150 ft, and 315 ft,

each slot having a total height of 72 ft. We have made four

different assumptions for the width, respectively 50 ft, 100 ft,

200 ft, and 400 ft. These widths should be contrasted with the

outflow diameters of 72 ft, and with the dimensions of the inflow

cone.

2-5

* The four widths, and the three horizontal slits, are

shown in Figures 2.3 and 2.4, which are otherwise identical to

Figures 2.1 and 2.2. The assumption is clearly a gross one.

Physically the warm inflow may be expected to be roughly axisymetric

! ' in the surface layers around the cone. The outflow Jets should

entrain fluid from the sides (Figure 2.4) and from above and below

(Figure 2.3). The model includes the vertical entrainment, but the

horizontal entrainment is lost in the azimuthal and statistical

averaging. Thus, the results can only be regarded as order of

magnitude estimates for the near flow.

As the figures show, the 50 ft width assumption gives an

outflow area smaller than the prototype circle (3600 sq. ft. as

compared with 4O71.5 sq. ft.). In addition, the horizontal

spreading and entrainment are not at all accurately modeled. Weregard this width as too small. Similarly, the assumption of a

width of 400 ft is extremely high. The quarter-circumference of a

circle with radius 233 ft is only 366 ft. Thus, if the four outflows

were combined into a circular slot of height 72 ft, the flow woald

still be stronger than in the 400 ft model. We believe that the

prototype flow is in a sense bracketed by the results obtained

assuming widths of 100 ft and 200 ft.

The vertical velocity profiles for the plant inflow andKoutflows are defined by equations (14) and (15) in Part I. The

function (1 - x ) was adopted to avoid the need to resolve

discontinuities; the use of a "top-hat" function, or the function(1 - x ) (to represent the horizontal average over a circular port),

might have been more appropriate. Again, this is an approximation,

and only order-of-magnitude accuracy can be expected. With a fixed

total flux at each port, the choice of this function does not affect

the results strongly.

2-6

FIGURE 2.3. NRFL02 Model of the

Lockheed Baseline OTPP Design

This figure is a repetition of Figure 2.1, with lines

added to indicate the horizontal slots used to model the external

flow for a single power module in a rectangular domain. The half-

module facing the reader is the one modeled. The warm inflow is

modeled by a slot of height 72 ft centered at 75 ft. The warm and

cold outflows are modeled by slots of height 72 ft, centered at

150 ft and 315 ft respectively. Four alternative widths for the

slots and the computational domain are used, with the half-widths

indicated in the figure as 25 ft, 50 ft, 100 ft, and 200 ft. The

depth of the computational domain is 500 ft. The left boundary in

the figure is a symmetry boundary. Figure 2.4 is a top view of the

same computational domain. For each assumed width, the flow speeds

are adjusted to give the correct total flux at the inflow and at

each outflow.

i-

I '

2-7

L2

114

50 10020

3 - .l I _ I

III I

I I

12-6

Ii Iji I

SI I

I I II I

I I I

I I

SL500 ~ --" - -- o

2-8

I

. ///I

\ \oo&\

/-- 50

IL - - -- -25

- -- - 25

--- - o50

/ 'I//

.00,I 200

FIGURE 2.4. Top View of the Lockheed Baseline OTPP DesignShoving the Four Alternstive Widths for the NHFL02 Model

of the Outflowe and Wars Inflow for One Module.

The rectangular couputational damln extendis to the rightof the power module with depth 500 ft and with the four alternative

half-vidths shown. The warm inflow sea nt and the two outflow, aremodeled by rectangular ports on the left-hand bomdary, occuVIng thewhole width, and with vertical profiles as indicated in Figures 2.3

and 3.9 through 3.12.

2-9

The IFL02 model of the Lockheed OTPP requires the

specification of the temperature, the downward velocity, and the

turbulent kinetic energy density, at the outflov ports. These

quantities are determined by equations (16) in Part I. The inflow

velocity is 10° below the horizontal for the cold inflow. The varm.

outflow temperature is 30F colder than the mean warm inflow

temperature. The cold outflow is at 450F, 2.4 0 F warmer than an

assumed cold inflow temperature of 42.6oF in the deep water. The

turbulent kinetic energy density is 0.3 times the square of the

maxium inflow speed at the port.

2.3 TIE NRFL02 TURBULENCE MODEL

The turbulence model used in NRFL02 is described indetail in Part I of this report. Briefly, it Is a four-parameter

first-order statistical closure model, with an imposed length scale,

and a single scalar turbulent diffusivity. The turbulent kinetic

energy density equation is standard, with generation, decay,

advection, and diffusion. The diffusivity determination includes

the suppression of vertical turbulent mixing by the density strati-

$, fication. An added term in the vertical equation of motion represents

the damping of vertical motion by turbulence and the coupling with

the random internal wave field.

The four parameters used in these computations were

chosen on the basis of stratified turbulent wake computations, and

have not been tuned to experiments on flows driven by inflows andoutflow jets. The values were

C.i L a 20 ft,

Scf a 0.5,

c * 0.1,8

0w .1.0.

2-10

iC

The results show a soewhat limited sensitivity to these parameters,

because the central effect is that the turbulent kinetic energy, and

therefore also the diffusivity, grow for strong shear and decay for

strong stratification. The relevant ratio is the Richardson number

Ri - N2/ (ui., + ui) "

cf. equation (13d) in Part I.

2.4. THE ASSUMED AMBIENT OCEAN CONDITIONS

The code NRFLO2 requires ambient profiles for the

temperature and the turbulent kinetic energy density, as described

in Parts I and III. The ambient temperature profile is given by

I equation (18a) of Part I, with

Tt - 61.5 ,

T - 1I.0°F,r

dt - 300 ft,

zt = 75 ft;

the profile is displayed in Figures 3.9 through 3.12. The surface

temperature is 800F, the temperature at 500 ft is 44-5F, and the

temerature at very great depths is 39.5°F.

c The ambient turbulent kinetic energy density profile is

given by equation (18c), vith

zE =125 ft,

SE~ t-8 2 23 Eo 7.4 x 10 f t2/sec

The results are insensitive to the ambient turbulence, provided it

is not unreasonably large.

2-31

The code NROL02 assumes zero ambient current. Calcula-

tion of the near-field flow for an operating OTPP in the presence of

an ambient relative current distribution requires a three-

dimensional code. We plan to develop such a code, NRFL03, in the

future.

I2

I

I f

2-12,-

* 3. RESULTS AND DISCUSSION

3.1. NUMERICAL RESULTS

* Our numerical results for the four models of the near-

field turbulent flow for the Lockheed baseline OTPP are displayed in

three figures for each case and in an accompanying table. The first

group of four figures is a blow-up of the stream lines in the lefthalf of the computational domain, together with a listing of the run

data, and the result for the mean temperature of the warm inflow.

The second group of four figures shows the isotherms and the stream

lines for the full computational domain. The third group of four

figures shows for each case the contours of the turbulent diffusivity,

and the profiles at the left and right boundaries of the temperature

and the horizontal motion.

The results for the four assumed region widths are

suimarized in Table 3.1. The table lists the maximum imposed speeds

at the inflow and outflows; these are inversely proportional to the

width. The assumed flow profiles are defined using the function

(1 - x2, as described in Part I.

The warm inflow temperatures are probably our most

important result. The extreme values, for widths 50 ft and 400 ft,

are 75.70F and 78.60. We believe that with the assumed ocean

9temperature profile, the OTPP baseline design would experience an

average inflow temperature between 77.20 and 77.90F; these are the

results for widths 100 ft and 200 ft.

The table also displays the warm outflow temperatures,S 30F cooler than the inflow temperatures. The cold outflow tempera-

tures are ilposed, at 45 F; this is 2.40F warmer than the assumed

cold inflow temperature of 42.601 from the deep water.'

3-1

FIGURE 3.1 Stream Lines near the

Plant for Width 50 Feet

The displayed domain is 500 ft deep and extends 375 ft

to the right of the OTPP outflows; this is the left half of the full

computational domain. The outflows are centered at depths 150 ft and

315 ft, with widths 72 ft. The stream lines are drawn at contour

intervals of 2000 cu ft/see. The flux of 15000 cu ft/sec leaving the

cold outflow (lines 1 through 7) falls till it finds its own level.

In doing so, it entrains 8000 cu ft/sec from the warm outflow

(lines 8 through 1). The remaining 4000 cu ft/sec of warm outflowC recirculates into the warm inflow (lines 2 and 3). The other

8000 cu ft/sec of warm inflow comes from the surface layers (lines 1,

0, 9 and 8). As a result of turbulent mixing and recirculation, in

this very strong flow computation, the mean plant inflow temperature

is 75.7°F.

The table at the bottom shows the data used for this

computation. This data is dimensionless, and is determined from the

dimensional data as described in Part III of this report.

3-2

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3-3

FIGURE 3.2. Stream Lines near the




computational domain. The outflows are centered at depths 150 ft

and 315 ft, with widths 72 ft. The stream lines are drawn at contour

intervals of 2000 cu ft/sec. The flux of 15000 cu ft/sec leaving the

cold outflow (lines 1 through 7) falls till it finds its own level.

In doing so, it merges with and entrains 10700 cu ft/sec from the

warm outflow (lines 8 through 2). The remaining 1300 cu ft/sec of

warm outflow recirculates into the warm inflow (line 3). The

remaining 10700 cu ft/sec of warm inflow comes from the surface

layers (lines 2 down to 8). As a result of the turbulent mixing and

recirculation, in this stzong flow computation, the mean plant

inflow temperature is 77.20F.


computation. This data is dimensionless, and is determined from

the dimensional data as described in Part III of this report.

3-4

- If

* 3-14

t V





computational domain. The outflows are centered at depths 150 ft

and 315 ft, with widths 72 ft. The stream lines are drawn at contour

intervals of 2000 cu ft/sec. The flux of 15000 cu ft/sec (lines 1through 7) leaving the cold outflow falls past its own level, and

rises again. In doing so, it entrains 2000 cu ft/sec from the warm

outflow (line 8) and 4000 cu ft/sec from larger distances (lines 9

through 0, coming in at depth 275 ft and leaving at 300 ft). Therest of the warm outflow (lines 9 and 0 through 3) flows out

horizontally, except that 300 cu ft/sec recirculate into the warminflow. The rest of the warm inflow comes in horizontally. As a

result of the weak recirculation, and the turbulent heat transport,

the mean inflow temperature is 77.90F.


computation. This data is dimensionless, and is determined from the

dimensional data as described in Part III of this report.

3-6

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The displayed domain is 500 ft deep and extends 375 ftto the right of the OTPP outflows; this is the left half of the fullcomputational domain. The outflows are centered at depths 150 ftand 315 ft, with widths 72 ft. The stream lines are drawn at contour

intervals of 3000 cu ft/sec. The flux of 15000 cu ft/sec (lines 1through 5) leaving the cold outflow falls till it finds its own

level. In doing so, it entrains 3000 cu ft/sec of deep cold water(line A) and 10000 cu ft/sec of water from about 280 ft (lines 6through 8). The warm outflow of 12000 cu ft/sec mostly flows outhorizontally (lines 6 through 8), entraining about 1000 cu ft/secboth above and below (lines 5 and 9). The recirculation flux isonly 100 cu ft/sec in this weak flow case (line 9). The mean inflowtemperature is 78.6 0 F. There is very little reduction due torecirculation or turbulent heat transport in this weak flow

computation.

The table at the bottom shows the data used for thiscomputation. This data is dimensionless, and is determined from thedimensional data as described in Part III of this report.

3-8

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3-9 .U I~0 a

FIGURE 3.5. Temperature Distribution and

Stream Lines for Width 50 Feet

The displayed domain is 500 ft by 750 ft, and is the

whole of the computational domain. The left half of the stream lines

figure is shown expanded in Figure 3.1. The temperature contours

labeled 6 through 9 and 0 through 6 correspond to multiples from 16

through 26 of the contour increment 30F, and thus, to temperatures

from 48°F to 780F. The stream lines are drawn at a contour increment

of 3000 cu ft/sec, and thus, show 15000 cu ft/sec leaving the cold

outflow, and a further 12000 cu ft/sec leaving the warm outflow and

entering the warm inflow. The surface is a 15000 cu ft/sec contour,

with the label 5.

The isotherms and stream lines are roughly horizontal

over most of the region. There are irregularities at the artificial

boundary on the right, where the flow is adjusted to find its own

density level, and there is evidence of some remaining fluctuations

in the form of internal waves. The temperature is not constant on

stream lines near the left boundary, because of the turbulent

diffusivity distribution shown in Figure 3.9. This effect is

pronounced near the warm inflow, where the 78°F contour crosses two

stream lines.

The flow pattern shown by the stream lines is discussed

in the caption to Figure 3.1.

3-10

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through 26 of the contour increment 30F, and thus, to temperatures

from 45 0F to 780F. The stream lines are drawn at a contour increment




with the label 5.







diffusivity distribution shown in Figure 3.10. Nevertheless, a

substantial part of the inflow water is at temperatures over 780 F,and the average inflow temperature is 77.2 0F.

The flow pattern shown by the stream lines is discussed

in the caption to Figure 3.2.

31', 3-12

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labeled 5 through 9 and 0 through 6 correspond to multiples from 15through 26of the contour increment 3,and thus, to temperaturest r u h 26 of t e c n o r i c e e t 3°F , an th , to e m r tu s

from 450F to 78 0F. The stream lines are drawn at a contour increment

of 5000 cu ft/sec, and thus, show 15000 cu ft/sec leaving the coldoutflow, and a further 12000 cu ft/sec leaving the warm outflow and

entering the warm inflow. The surface is a 15000 cu ft/sec contour,with the label 3.

The isotherms and stream lines are roughly horizontalover most of the region. There are irregularities at the artificialboundary on the right, where the flow is adjusted to find its owndensity level, and there is evidence of some remaining fluctuations



diffusivity distribution shown in Figure 3.11. The 780 F contour

reaches the left boundary at 80 ft, and the mean inflow temperature

is 77.9°F. A strong thermocline separates the warm outflow from the

cold outflow.

The flow details, and the entrainment from infinity, are

discussed in the Figure 3.3 caption.

3-114

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through 26 of the contour increment 3°F, and thus, to temperatures

from 450F to 780 F. The stream lines are drawn at a contour increment




with the label 3.





in the form of internal waves. The temperature is practically

constant on stream lines near the left boundary, because of the

small turbulent diffusivity distribution shown in Figure 3.12. The

780 F isotherm crosses the left boundary at 90 ft, and the mean inflow

temperature is 78.6 0 F.

The stream lines are discussed in the Figure 3.4 caption.

There is substantial entrainment above and below both outflow jets.

3-16

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c 3-1T

FIGURE 3.9. Diffusivity Distribution, and

Boundary Profiles of Horizontal Flow and

Temperature, for Width 50 Feet

4 . The turbulent diffusivity is displayed in a domain 500 ft2by 750 ft, with a contour increment of 15 ft /sec. The contour

labels are multiples of this increment. The diffusivity reaches its

maximm of 85.5 ft2 /sec at the cold outflow, and is only slightly

smaller at the warm inflow and outflow. The minimum value on the2left boundary is 40.1 ft /sec, and it drops rapidly with distance

from the plant, at all depths.

In the lower figure the scales for the horizontal flow

U and the temperature T and ambient temperature A, are as shown.

The ambient temperature increases from 44.50F to 800F, as described

in Section 2.4. The flow profile on the left boundary is imposed

to represent the warm inflow at 75 ft and the warm and cold outflows

at 150 ft and 315 ft; the maximum speeds are 6.38 ft/sec, 6.38 ft/sec,

and 8.50 ft/sec. The flow profile on the right is towards the plant

down to 160 ft; most of this flux eventually enters the warm inflow.

The temperature on the left boundary is 45 F at the cold

outflow, and 72.70F at the warm outflow. This is 30F cooler than the

average inflow temperature. The temperature at the left boundary issubstantially below the ambient temperature. The flow is adjusted

near the right boundary so that the outflow finds its own density

level; this adjustment was successful here except that there was no

water to flow out at below 50 0 F, because of the turbulent mixing.

The horizontal flow in the lower right figure determines

the far-field environmental impact. Water is added to the ambient

ocean in certain temperature and density ranges, and is withdrawn from

the ambient ocean in others. With very large scale OTPP operations,

the thermocline structure and current system will be modified.

3-18




The turbulent diffusivity is displayed in a domain 500 ft

by 7 50 ft, with a contour increment of 7.5 ft 2/sec. The contour

labels are multiples of this increment. The diffusivity reaches its

maximum of 42.9 ft 2 /sec at the cold outflow, and is only slightly

smaller at the warm inflow and outflow. The minimum value on the

left boundary is 18.7 ft2/sec, and it drops rapidly with distance

from the plant, at all depths.

In the lower figure the scales for the horizontal flow U

and the temperature T and ambient temperature A, are as shown. The

ambient temperature increases from 44.50F to 800F, as described in

Section 2.4. The flow profile on the left boundary is i e to

represent warm inflow at 75 ft and the waz. ,a - Outflows at

150 ft and 315 ft; the maximum speeds are 3.19 ft/sec, 3.19 ft/sec,


down to 150 ft; most of this flux eventually enters the warm inflow,

but there is some entrainment by the warm outflow Jet leaving at

200 ft. The cold outflow leaves between depths of 300 ft and 450 ft,I-and there are small inflow regions above and below, due to entrainment.

The temperature on the left boundary is 45 0F at the cold

outflow and 74.2°F at the warm outflow (3 0 F cooler than the average

of the warm inflow). The temperature on the right agrees well with

the ambiwt temperature distribution. -





*the thermocline structure and current system will be modified.

3-20"

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by 750 ft, with a contour increment of 3 ft 2/sec. The contour labels

represent multiples of this increment. The diffusivity reaches its

maximum of 21.7 ft /sec at the cold inflow, and is only slightly

smaller at the warm outflow. The minimum value on the left boundary

is 7.1 ft /sec, and it drops rapidly with distance from the plant,

at all depths.

In the lover figure the scales for the horizontal flow U

and the temperature T and the ambient temperature A, are as shown.

The ambient temperature increases from 44.5°F to 80°F, as described

in Section 2.4. The flow profile on the left boundary is imposed,

to represent the warm inflow at 75 ft and the warm and cold outflows

at 150 ft and 315 ft; the maximum speeds are 1.60 ft/sec, 1.60 ft,


down to 120 ft; some of this flux eventually enters the warm inflow

and some is entrained. The warm and cold outflows at 190 ft and

370 ft have inflow regions, above and below, due to entrainment. The

flow has been adjusted successfully so that the temperature distribu-

tion agrees with the ambient temperature.

The temperature on the left boundary is 450 F and 74.9°F

at the outflows. At the inflow the temperature is close to the

ambient temperature, with average value TT.9 F.





the thermocline structure and current system will be modified.

3-22

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by 750 ft, with a contour increment of 2 ft 2/sec. The contour labels

represent multiples of this increment. The diffusivity reaches its

maximum of 13.5 ft /sec at the cold outflow, and is only slightly

smaller at the warm inflow and outflow. The minimum value on the2

left boundary is 0.4 ft /sec, and it drops rapidly with distance from

the plant, at all depths. Thus, there is effectively no turbulent

cooling of the surface layers by the cold outflow.

In the lower figure the scales for the horizontal flow U

and the temperature T and ambient temperature A, are as shown. The

ambient temperature increases from 44.5 0F to 800F, as described in

Section 2.4. The flow profile on the left boundary is imposed, to

represent the warm inflow at 75 ft and the warm and cold outflows at

150 ft and 315 ft; the maximum speeds are 0.80 ft/sec, 0.80 ft/sec,

and 1.06 ft/sec. The flow profile on the right is complicated; the

warm and cold outflow jets, at depths 185 ft and 370 ft, each have

inflow regions above and below due to entrainment. The inflow

temperature profile on the left is very close to the ambient

temperature, and the mean plant inflow temperature is 78.6 0 F. The

warm outflow temperature is 75.6 0 F (30F cooler); the cold outflow is

at 450F, (2.40F warmer than the deep cold inflow).

The horizontal flow in the lover right figure detezuines


ocean in certain temperature and density ranges, and in withdram from

the sabient ocean n others. With very large scale OTPP operations,

the thermocline structure and current system vill be modified.

3-240

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3-26

The Vwz inflow of 12000 cu. ft/sec is partly verm surfaceU flux, coming in horizontally from the surface layers of the ocean.

The other part is recirculated water from the warm out flow. The

table displays these two fluxes, totaling to 12000 au ft/sec In each

case. For widths 50 ft and 100 ft, the recirculation flux is

substantial; for widths 200 ft and 400 ft, it is very small.

The temperature distribution at the warm inflow is lower

than the ambient temperature distribution for two main reasons. The

first reason is recirculation of warm outflow water (after cooling

by 3°F in the evaporators); the temperature loss is proportional to

the recirculation flux. The second reason is turbulent ixing with

the warm outflow and with the cold outflow. We therefore present in

the table the maximum turbulent diffusivity for the four cases.

This maximum is reached in the cold outflow. The values near the

warm inflow and outflow are only fractionally smaller in each case

(see Figures 3.9 through 3.12). But the minimum value on the

boundary between the outflows, is very much smaller for width 400 ft.

Thus, there is significant turbulent heat transfer between the cold

outflow and the surface only for the smaller widths. The table,

therefore, displays the minimum value of the turbulent diffusivity

on the left boundary between the outflows.

3.2. VALIDITY OF THE RESULTS

The validity of the results depends on two basic

requirements:

0 the numerical results are a good solution of themodel problem as a system of equations; and

e the model problem is a reasonable approximation tothe prototype flow conditions.

These requirements are discussed in the following two subsections.

3-27

0

3.2.1. Validity of the Nuerical Results

The validity of the numerical results requires:

* no major coding errors;

0 sufficient spatial resolution in the finitedifference representations; and

• minimum error due to the remaining minorfluctuations in the solution.

First, the code is listed and documented in Part III of

this report, end the different routines have been carefully analyzed.

We believe the code is a faithful implementation of the equations and

boundary conditions and of the numerical representations described in

Part I. Although the code is fairly long, a substantial part is

concerned with output, and the remainder is therefore reasonably

easy to verify.

Secondly, our representation of the equations is second

order in the mesh spacing. Our solutions displayed in Figures 3.1through 3.12 do not show any significant variation on length scales

smaller than a few mesh intervals. And in developing the code

URFL02, we tested carefully for dependence of the numerical results

on resolution in simpler problems.

Thirdly, the numerical results for these four cases are

slightly unsatisfactory because we have not completely eliminated

the residual fluctuations in time. We are applying Newtonian damping

to the vertical motion near the right boundary, together with a

pressure boundary condition designed to ensure that fluid leaves the

region at a depth agreeing with its temperature. These methods have

already reduced the amplitude of the residual fluctuations to a few

percent, in terms of the quantities displayed in Table 3.1. We

believe that our system of equations bas steady solutions, and ve

plan to develop numerical methods to eliminate the present

I. fluctuations completely.

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3.2.2. Validity of the Model

The validity of our model as a description of the

prototype flow near the Lockheed baseline OTPP in a real ocean

environment depends on the following two requirements:

0 validity of the statistical turbulence model; and

0 validity of the two-dimensional approximation tothe Lockheed OTPP.

Neither requirement is known to be met at present. We discuss the

requirements below.

Our turbulence model and parameters were chosen on the

basis of our experience in stratified turbulence modeling for wake

calculations, but its validity is doubtful for the present applica-

tion. There is only one turbulent diffusivity, the same for each

variable and for horizontal and vertical diffusion. An arbitrarily

assigned constant length scale is used in the model. And the

coefficients have not yet been tuned to the results of stratified

turbulence experiments with inflows and outflows.

we plan to upgrade the turbulence model in the future.

We will use a tvo-equation first-order statistical closure model,

with an additional empirical equation to determine the turbulence

length scale. The turbulent diffusivities will be tensors, and will

use different coefficients for each variable. The effect of density

stratification in impeding vertical turbulent transport will be

included as in the present model. There will be a large number of

unknown dimensionless parameters in the model. We plan to determine

them by tuning the results against the best available experimental

measurements for the statistics of stratified turbulent motions

driven by inflows and outflow.

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The two-dimensional MUPL2 model of the three-dimensional

flov near the prototype Lockheed baseline OTPP was described in

Section 2.2. As previously stated, the approximation is substantial.

The results cannot, therefore, be relied on in detail, though they

are valid in order of magnitude. In a real sense, this is a baseline

external flow study.

3.3. CONCLUSIONS

We draw the following conclusions from the research

presented here:

1. OTPP operation in the absence of an ambient current

does not raise insuperable recirculation problems. Because of the

stratification, the inflows and outflows are essentially horizontal

at large distances from the plant.

2. Our best estimate for the warm inflow temperature

for the Lockheed OTPP, operating with the ambient temperature profile

described in Section 2.4. and displayed above, is TT.5 0 F. This is

2.50F below the surface temperature. Our extreme estimates were

75.7TF and 78.60F. Changes in design, with wider, slower inflows

and outflows, and with the warm inflow nearer the surface, could

probably increase the warm inflow temperature up to almost the

surface temperature.

3. Our numerical methods have been successfully

applied in this relatively simple two-dimensional code, and we do

not anticipate any insuperable problems in extending them to

amisymmetric and three-dimensional codes.

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ACKNOWLEDGEMENTS

This report was prepared as part of the research program,

"Theoretical Fluid Dynamical Studies of Resource Availability and

Environmental Impact of Ocean Thermal Power Plants", in the Naval

Research Laboratory, supported by the Ocean Thermal Energy Conversion

Program, Division of Solar Energy, Energy Research and Developnent

Administration, under ERDA contract E (49-26) 1005. The majority of

the work was performed under the subcontract N00014-76-C-0610 with

Science Applications, Inc., and this constitutes the Final Report on

that contract.

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REFERENCES

1. Douglass, R. (1975), "Ocean Thermal Energy Conversion Researchon an Engineering Evaluation and Test Program", Vols 1-5, TRW,Redondo Beach, Calif.

2. Piacsek, S.A., Martin, P.J., Toomre, J., Roberts, G.O. (1976),"Recirculation and Thermocline Perturbations from Ocean ThermalPower Plants", Report NRL-GFD/OTEC 2-76, Geophysical SimulationSection, Code 7750, Naval Research Laboratory, Washington, D.C.

3. Piacsek, S.A., Toomre, J., Roberts, G.O. (1975), "GeophysicalFluid Dynamics Background for Ocean Thermal Power Plants",Report NRL-GFD/OTEC 11-75, Geophysical Simulation Section,Code 7750, Naval Research Laboratory, Washington, D.C.

4. Roberts, G.O., Piacsek, S.A., Toomre, J., "Two-DimensionalNumerical Model of the Near-Field Flow for an Ocean ThermalPower Plant, Part I. The Theoretical Approach and aLaboratory Simulation", NRL-GFD/OTEC 5-76, Science Applications,Inc. and Geophysical Simulation Section, Code 7T50, NavalResearch Laboratory.

5. Roberts, G.O., Piacsek, S.A., Toomre, J., "Two-DimensionalNumerical Model of the Near-Field Flow for an Ocean ThermalPower Plant, Part III. Manual for the Computer Code NRFLO2",NRL-GFD/OTEC 7-76, Science Applications, Inc. and GeophysicalSimulation Section, Code 7750, Naval Research Laboratory.

6. Triable, L.C. (1975), "Ocean Thermal Energy Conversion PowerPlant Technical and Economic Feasibility", Lockheed Missileand Space Co., Inc., Report NSF/RANN/SE/GI-C937/FR/75/I,Sunnyvale, Calif.

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