Turbulent diffusion from a point source, laboratory data

Turbulent diffusion from a point source, laboratory data

~mcupkric EnviroMvnr Vol IS No IO, pp 1969-2002 1984

STRATIFIED AND NEUTRAL FLOWS AROUND A THREE-

OF SURFACE CONCENTRATIONS

Meteorology and Assessment Division, Environmental Sciences Research Laboratory, U.S Environmental

Protection Agency, Research Triangle Park, N.C 27711, U.S.A

and

J c R khWt Department of Applied Mathematics and Theoretical Physics, University of Cambridge,

Cambridge CB3 9EW, U.K

(First received 5 September 1983 and received for publication 19 March 1984)

Abstract-Towing-tank and wind-tunnel measurements of the concentration distributions on the surface of

a hill when a plume impinges from an upwind source are presented The stability is varied between very stable and neutral The results are compared with the theories developed in Part I When the source is below the dividing-streamline height H, the plumes impact on the front surface of the hill, yielding surface concentrations nearly the same as would be observed at the plume centerline in the absence of the hill However, eddying in the wake can cause oscillations in the plume upwind so as to increase the area of impingement and decrease the average concentration When the source is above H, the plume surmounts the hill top, but if it is only slightly above H, maximum surface concentrations can again essentially equal those that would be observed at the plume centerline in the absence of the hill The maximum surface concentration

decreases very rapidly with further increases in source height The location and value of the maximum surface concentrations are found to he extremely sensitive to slight dispkuzments of the source from the stagnation streamline when the source is below H, The general assumptions of the potential flow models developed in Part I to provide estimates of surface concentrations on three-dimensional hills are useful

Key word index: Turbulent diffusion, stratified flow, wind tunnel, towing tank, complex terrain, air pollution

NOMENCLATURE constant or area

constant

concentration

maximum surface concentration

plume centerline concentration in the absence of the

hill

source diameter

constant

Froude number, U/Nh

acceleration due to gravity

hill height

dividing-streamline height

source height

plume width (standard deviation)

half-length of hill (at half-height)

coordinate normal to plume axis (see Part I)

distance normal to plume axis from centerline to hill

surface (see Part I)

*On assignment from the National Oceanic and u;

Atmospheric Administration, U.S Department of

distance along streamline from source to hill top surface coordinate (arc distance along hill surface from top center of hill)

location of maximum concentration towing speed or mean velocity free-stream wind speed or towing speed streamwise coordinate

source coordinate in x-direction offset of source from centerline in lateral direction vertical coordinate

vertical deflection of streamline from far-upstream elevation

lateral displacement from hill centerline of plume impingement point

azimuthal coordinate perpendicular to s and II (see Part I)

angular coordinate (zero in positive x-direction) fluid density

plume width (standard deviation) normal to plume axis

standard deviation of lateral concentration distri- bution in absence of hill

standard deviation of vertical concentration distri- bution in absence of hill

H and J.C R

plume width (standard deviation) in azimuthal

direction

along-surface plume width defined as distance be-

tween points where concentration is l/l0 of maximum

value

lateral plume width defined as distance between points

where concentration is l/l0 of maximum vahte in

absence of hill

vertical plume width defined as distance between

points where concentration is l/l0 of maximum value

vertical plume width defined as distance between

points where concentration is l/IO of maximum value

maximum nondimensjonai plume centerline concen-

tration in absence of hilf

maximum nondimensional plume centerline concen-

tration in absence of hill assuming a best-fit Gaussian

approximation to the measured distribution

nondimensional concentration in wake of hill

1 INTRODUCTION

The structure of strongly stratified flow over a three-

dimensional hill may be usefully envisioned as com-

posed of two layers: a lower layer of essentially

horizontal flow wherein plumes from upwind sources

impinge directly on the hill surface, and an upper layer

wherein pIumes from upwind sources may pass over

the hill top This flow structure was suggested by

theoretical arguments of Drazin (1961) and dem-

onstrated through laboratory experiments by Riley et

al (1976) Brighton (1978) and Hunt and Snyder (1980)

(hereafter referred to as HS) HS performed sah-water-

stratified towing-tank studies using a simple belf-

shaped hill with a uniform velocity profiie and a linear

density gradient in the flow approaching the hill They

showed that, under these conditions, the depth of the

lower layer is predictable as

where H, is the dividing-streamline height (depth of

lower layer), h is the hill height, and F = U/N/t is the

Froude number based on the uniform upstream

velocity U and Brunt-Vat&a frequency N Snyder et

al ( 1980) demonstrated that ( 1) was applicable to other

shapes of axisymmetric hills and presented another

simple formula and supporting experimental data for

determining whether an elevated (step) inversion

would surmount a hill Snyder et al (1984) derived a

more general integral formula for predicting the

dividing-streamline height for arbitrary shapes of wind

profiles and stable density gradients This formula was,

in fact, suggested much earlier in a note by Sheppard

(1956) based on kinetjc/potential~nergy exchange

arguments

f&(H,)=g

h (h - 2) (- dp/dz)dz, (2)

“n

where pUZ, is to be evaluated far upstream at the elevation H,, g is the gravitational acceleration and dp/d: is the vertical potential density gradient This formula easily reduces to the simpler formulas dis- cussed above using the boundary conditions ap- plicable to those cases

Baines (1979) and Weil et al (1981) confirmed the

two-layer concept for strongly stratified flow, but suggested the formula

H,;h= I-2F (3)

for barriers with very small gaps Snyder et al ( 1984) however, cast doubt upon the results of Baines and Weil ez al by showing evidence suggesting that steady- state conditions had not been established in their experiments Further, Snyder et al confirmed the general validity of (2) through additional laboratory studies using a wide range of hill shapes, density profile shapes and strong shear flows

The utility of the two-layer concept arises because the transport and diffusion in each layer may be analyzed quite separately and independently of one

another using different but we&established techniques

in each layer

In the lower layer, for sufficientty strong stratifi- cation (F d I), the flow is approximately horizontal Outside the wake, the velocity field at height i < Ho is the same as two-dimensional potential flow about a cylinder The shape of the cylinder is defined by the contour of the hill at height 2 As in neutral flow, turbulence does not affect the mean flow outside the wake or the thin boundary layer on the hill surface Given this mean flow field and the fact that stable stratification limits vertical diffusion, the calculation of dispersion from a point source at height H, (below the dividing streamline) is the same as that from a line source near a cylinder (See Fig 1 and Hunt ef al., 1979, and Weil er al., 1981)

In the upper layer, buoyancy and inertial forces control the flow 3s it passes over the hill It has been suggested (e.8 Rowe et al., 1982; Bass et al., 1981) that this upper layer flow is approximately potential flow Although this is not theoretically correct, in this paper

we examine whether this hypothesis can lead to useful estimates for diffusion in the upper layer

Potential flow modeling, if valid, is highly useful because of its simplicity, adaptability and flexibility Streamline patterns are calcufated from potential-flow theory, and the di~usion along those streamline paths can therefore be calculated using different techniques For example, in Part I, plume widths and concen- trations were estimated by assuming constant dif- fusivities, while Bass rt ul (1981) used Gaussian plume assumptions with the Pasquilf-Gifford curves (Turner, 1970) Further, since any streamline found from poten- tial flow caiculations can itself be considered a hill surface, the technique can be easily adapted to wide ranges of hill shapes For example, Bass et al have

constructed an algorithm for calculating concen-

Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill-H 1971

( a) Stde view of plume implnglng on hilt ot Low Froude number

f b) Section through hilt, plume and woke at z = Ii,

Fig 1 Simplification of plume impingement on threedimensional hill under very stable

stratification

trations on the surface of a series of 20 bluffs whose

maximum slopes range from 8 to 72”

The primary purpose of the laboratory experiments

to be described in this paper was to test the predictions

of the theory developed in Part I, i.e with simple

mathematical models that are based on straightfor-

ward physical hypotheses of the flow patterns and

turbulent diffusion around simple shapes of hills The

specific aims of the experiments were given in Part I,

but are repeated here for the sake of completeness

(a) To investigate the two rival hypotheses about

the maximum surface concentration C, in stably

stratified flow; these are (i) the EPA valley model (Burt

and Siater, 1977), where, effectively, C, is assumed to

be twice the centerline ~on~nt~tion C” of the plume in

the absence of the hill, and (ii) the model of Hunt and

Muihearn (1973), where C, isapproximately equal to

C”

(b) To measure the effects of small lateral dispiace-

ments of the source from the centerline of the hill This

is particularly important for understanding the effects

of low frequency wind direction fluctuations (plume

meander) on the average surface concentration (see

Appendix)

(c) To measure the dist~bution and maximum

values of surface concentrations on a three-

dimensional hill in neutral and stably stratified flow

and to compare these with those on level ground and

on two-dimensional hills

In these experiments, three primary parameters were varied: the height of the source, the lateral offset of the source from the centerline, and the Froude number, As

in HS, stably-stratified flow experiments were con- ducted in the at-water-stratify towing tank and neutral flow studies were done in the wind tunnel Neutrally buoyant dye was used as the source efluent

in the towing tank and ethylene in the wind tunnel The ex~rimen~i results presented herein are primarily the surface concentration measurements as functions of the above three parameters

Note that in these experiments no upwind turbu- lence is generated, so that outside the wake, diffusion occurs only because of turbulence created at the release point, i.e the model stack Since in very stable con- ditions, the diffusion of chimney plumes is largely determined by initial mixing within the plume (Weil, 1983), the present experiments are a reasonable simu- lation of such conditions In neutral conditions, our experiment is not a good simulation of the full-scale adiabatic boundary layer, but we include results ob- tained in the neutral, low-turbulence, uniform-flow wind tunnel for comparison with the strongly stable conditions

Within and near the wake, turbuhmce is generated that has significant effects on the diffusion, but even these effects are somewhat different when upwind turbulence is present Complementary studies of dflu- sion in neutrally stratified turbulent flows over hills of

1972 WILLIAM H SNYDER and J C R HUN.T

varying aspect ratio (crosswind width to height) have

been performed by Snyder and Britter (1984) and

Castro and Snyder (1982) Further studies of diffusion

over hills of varying streamwise aspect ratio (or slope)

have been performed by Khurshudyan et al (1981)

2 EXPERIMENTAL APPARATUS AND TECHNIQUFS

Most of the detaik of the experimental apparatus and

techniques were given in HS and in a laboratory report by

Hunt et al (1978) (hereafter referred to as HSL), so that only a

brief review is given here Special techniques used for the

measurement of concentrations are presented in detail

A fourth-order polynomial hill [z(r) = h/(1 $ (r/L)4)] of

height 22.9cm was used in a stratified towing tank and in a

neutral wind tunnel The towing tank was 1.2m in depth,

2.4m in width and 25m in length and was stably stratified

with layered mixtures of salt water The effluent used was blue

food dye diluted with sufficient salt water to produce a plume

that was neutrally buoyant at the source height (typically, 1

part dye and 15 parts salt water) This dye mixture was

emitted isokinetically from a bent-over ‘stack’ ofO635cm o.d

The stack exit was located 84.8 cm (3.7 h) upstream of the hill

center Twenty eight sampling ports were fixed on the hill

surface along each of the radial lines 8 = 180, - 16.5, - 90 and

0” (see Fig 7 of HSL) These sampling tubes protruded

2Smm above the surface so that measured concentrations

were not affected by molecular diffusion through a viscous

subiayer on the hill surface, Horizontal and vertical rakes of

tubes were also employed to obtain concentration profiles in

the wake (62cm downwind of the hill center) and in the

absence of the hill (at x = 0)

The wind tunnel had test section dimensions of 3.7m

x 2.1 m x 18.3 m (Snyder, 1979) The model was placed close

to the entrance to the test section in order to obtain a uniform,

non-turbulent flow over the model (purposely avoiding a

thick boundary layer), so that the resuhs obtained in the

neutral wind tunnel could be dire&y referenced to those

obtained in the stably stratified towing tank A mixture of air

and ethylene (C,H,) was emitted as the tracer gas from the

stack Since the molecular weight of ethylene is close to that of

air and since the volume fraction of ethylene in the eflhtent

was only about 6 3, the effluent was essentially non-buoyant

As the effluent release was isokinetic (0, = 3ms-‘), the

effluent Reynolds number was approximately 1000

2.1 Concentration measurements in the towing tank

For quantitative determination of concentrations, during

the tows samples were drawn through the ports on the hill

surface and through the sampling rake positioned

downstream of the hill The rake conststed of a hO~lZOIillti arm with tube spacings of Scm and a vertical arm with tube

spacings of 2cm A vacuum system shown schemancally m Fig 2, was used to withdraw approximately IOOcm’ 01 sample through each tube on the rake and on the hill surtacc into individual bottles

The concentrations of dye in these samples were then analyzed on a Bausch and Lomb Spectronic 20 calorimeter

‘Standards’ of known concentration were analyzed to estab- lish a calibration curve for the coiorimeter, and a numertcal scheme was used to interpolate between calibration points fsee below) The wavelength used on the coIortmrter was 505nm The output of the coiorimeter was monitored with a digital voltmeter, with care being exercised bycheckingand, ii necessary, readjusting the ‘zero absorbance’ between samples This technique permitted the measurement of concentrations

as small as 0.001 “, dye, or, since the source concentration was approximately 6 9,, it permitted the measurement ofdilutions between 6 and 6000 Smaller dilutions were measured by using less dye in the e&tent and larger dilutions (to perhaps 50,000) by using more dye

A numerical technique was used to interpolate between (and infrequently to extrapolate slightly beyond) the stan- dards Theoretically, the response of the coiorimeter should follow Beer’s Law, but, in practice, Beer’s Law was found not

to fit the standards very well Hence, a mod&d form of Beer’s Law using three constants (Y = exp (A f BC + DC’), where C

is thecoiorimeter output voltage, C isconcentration and A, 3 and D are constants to be determined) was used for each of two ranges of voltage The three constants for each range as well as the number of points covered by each were determined through minimizing the mean-squared error The two curves were then smoothly combined in the range of overlap, with a typical fit shown in Fig 3 Once the equation of the curve was established, it wasa simple matter, given the output voltage of the coiorimeter for a sample of unknown concentration, to

determine the concentration of the sample

One problem with the dye was of course, that II left a residual in the tank so that the background would normally increase with every tow The dye was controlled by the addition of small amounts of chlorine bleaching agent The water, before being placed in the tank, was pH-balanced and chlorinated (i-3 ppm), The chlorine bleaching process was rather slow-acting, so that over a period of an hour or so, it bleached-out any background residual in the towing tank A drop of sodium thiosuifate (antichior) in the sample test tubes, however, neutralized the chlorine in the samples and prevented bleaching, hence deterioration, of the samples collected The dye itself, of course, neutralized the chlorine during the bleaching process so that eventually the back- ground built up, but generally, several tows were possible at each stack height before the background dye had built up to the point where the water was no longer usable

Test tubes for collecting dye Efftuent reservoir

Verticoi and twirontol mke Fig 2 Schematic diagram showing system for collecting in towing tank

Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill II 1973

2.2 Concentration measurements in the wind tunnel

Concentration profiles were obtained in the wind tunnel by

collecting samples through a 1.6mm tube that was stepped

through the plume and analyzing them with a Beckman

Model 400 Hydrocarbon Analyzer (a Same ionization detec-

tor) in the continuous operating mode A 2-mht sampling time

was found to yield reasonably stable values of concentration

The aero and span were adjusted on the HC analyzer by using

‘zero’ air (less than 1 ppm total HCs) and a 1 y0 mixture of

methane in air, respectively The ‘zero’ and ‘span’ were

rechecked after each profile was measured to be certain that

they had not shifted previous tests run on the HC analyzer

showed its response to be linear with concentration, and that

it was 1.266 times as sensitive to ethylene as it was to methane

Surface concentration profiles were obtained by connect-

ing the tubing from the HC analyzer sequentially to each of

the surface ports (see Fig 4)

3 PRESENTATION AND DRTCURRION OF RESULTS

3.1 Concentration distributions in the absence of the

hill

Because it was of interest to compare concentrations

measured on the hill surface with those on the

centerline of the plume in the absence of the hill, it was essential to establish baseline conditions The baseline measurements in the towing tank were all made with the same density gradient (N = 1.33 rad s- ‘), at the same three speeds at which the hill was towed (6, 12 and 3Ocm s-l), at the source heights (H, = 3,9, 12.5 and 27 cm), corresponding to H,/h = 0.13, 0.39, 0.55 and 1.18, respectively Baseline measurements in the wind tunnel were also made under similar circum- stances in the absence of the hill The baseline concen- trations in the towing tank were measured at a point corresponding to the center of the hill (x = 0), 84.8 cm downstream of the source (x,/h = - 3.7) In the wind tunnel, additional measurements were made upstream and downstream of that point All nondimensional results are recorded here using the hill height h as

the length scale for convenient comparison with later results, e.g concentrations are normalized as x

= CCJ,h’/Q

Figure 5 shows typical behavior of plumes emitted above the surface boundary layer under strong stratifi- cation (F = 0.2); the horizontal plume width increased steadily with distance from the source, but the vertical plume width first increased due to the turbulence in the effluent issuing from the stack, then slightly decreased due to the strong stratification suppressing this turbu- lence and creating a gravity current (see later discus- sion) Plotted on the graphs with the data points are the

‘best fit’ Gaussian curves For the lateral profile, the fit

is quite good The spacing of the rake tubes was not sufficiently fine to resolve much detail in the vertical profile Note that the three points departing signifi- cantly from the best-fit Gaussian curve represent concentrations less than 0.5 % of the maximum in the plume and, in this case, represent the lower limit of resolution in the concentration measurement system, i.e background noise’

Plumes emitted above the surface boundary layer under a weaker stratification (F = l.O), showed a growth in horizontal width that was not as rapid as for the low Froude number plumes; the growth in vertical

detector Fig 4 Concentration measurements in wind tunnel (dimensions in cm)

1974 WILLIAM H SNYDER and J C R HUNI

~ 0

Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill II 1975

width, however, was small but positive

Figure 6 shows the plume behavior at F = 0.4, where

the stack exit was very close to the top of the boundary

layer (see later discussion) The growth of the horizon-

tal and vertical plume widths were quite rapid because

of the turbulence in the boundary layer; the plume

occasionally diffused to the surface At a Froude

number of 1.0 (Fig 7), the lateral spread of the plume

was about the same as that at the stronger stratification

(F = 0.4), but the vertical spread was about twice as

large, with the surface concentration being a significant

fraction of the maximum concentration aloft

Figures 8 and 9 show the concentration distri-

butions measured in the neutral stratification of the

wind tunnel (F = co) at two stack heights and at three

locations downwind of the stack The vertical and

horizontal widths of the elevated plumes were quite

comparable to each other, but much smaller than those

from the lower stack because the taller sources were

outside the boundary layer The vertical profiles of the

elevated plumes were slightly nonsymmetrical, show-

ing a slightly greater downward spreading which is

most likely dueto the turbulence generated in the wake

of the stack tube

The plume characteristics are summarized in Table 1

and Fig 10 The table is divided into two parts:

(a) H,/h = 0.13 and (b) H,/h L 0.39, because the high-

level plumes were quite similar to each other, yet quite

different from the low-level plumes Visual obser-

vations, some of which are shown in the photographs of

Figs 5-7, indicated that at F = 0.2, the boundary layer

was about 0.13 h thick; it thickened to about 0.22 h at

F = 1 and to about 0.28 hat F = 00 (see HSL, Fig 17)

Consequently, when H,/h > 0.39, the plumes were

above the baseplate boundary layer, and when H,/h

= 0.13, they were within it A rough estimate suggests

that, if this boundary layer were turbulent (from visual

observations, the boundary layer was obviously turbu-

lent), its thickness would be 6/h z 0.2 Since the gross

Richardson number of the boundary layer was about

0.07, there was only a small reduction of the turbulence

and the thickness of the boundary layer, so that this

crude estimate agrees quite well with the visual

observations

The plume widths for H,/h Z 0.39 are plotted in

Fig 10(a) We observe that as U, increases (or the

stratification decreases), Zy” decreases and Z,” increases

(X is the plume width, defined as the distance between

points where the concentration is one-tenth of the

centerline value, as shown in Fig 10(a); for a Gaussian

distribution, Z = 4.3~) There are two major factors

affecting this trend First, the initial plume size is equal

to the stack diameter (d = 6 mm = 0.026 h), so that the

density of the plume above its centerline is greater than

the surroundings and the density below less; hence, the

plume collapses vertically and spreads laterally as a

gravity current A simple estimate (see Wu, 1969)

suggests that due to this mechanism

Figure 10(b) shows the variation of the centerline concentration 1; As the stratification decreased (the speed increased), the turbulence in the effluent was initially stronger, it was suppressed less, and the dilution increased

3.2 Concentration distributions on and around the hill 3.2.1 General observations offlay structure and plume behavior Figure 11 presents several photographs of plumes emitted from stacks of various elevations upwind of the hill in flows of various Froude numbers The photographs are organized into three groups: (a) small Froude number with the effluent released below the dividing-streamline height [F < 1 and H,

< Ho = h(1 -F)], (b) small Froude number with effluent released at the dividing-streamline height (F

c 1) and H, = Ho) and (c) all Froude numbers with eflluent released above the dividing-streamline height (H, > Ho)

The main characteristic of the plumes emitted below

Ho is that they impinge on the hill surface, split, and travel round the sides of the hill [Fig 11 (a)] Upwind, the plumes are largely constrained to move in horizon- tal planes and vertical diffusion is severely limited (One exception is the plume emitted at H,/h = 0.2, Ho/h = 0.4, where it was obviously emitted into a turbulent boundary layer; the existence and depth of this turbulent boundary layer probably resulted from a near-neutral density gradient near the water surface This series of photographs was taken primarily for instructive purposes and the linearity of the density gradient was not as carefully maintained as was done in the remainder of the experiments.) The plumes were frequently rolled-up within an upwind vortex as they impinged on the hill surface This behavior did not appear to be regular or steady, nor predictable, but the

‘diameter* of the vortices (i.e vertical excursions of fluid parcels) appeared to be limited to approximately

Fh The vertical plume dimensions increased suddenly and substantially at the upwind stagnation point, whether due to the roll-up in the vortex or due to the vertical divergence of streamlines These vertical widths were roughly maintained as the plumes were swept around the hill surface The plumes lost elev- ation as they were transported around the sides of the hill Plumes that were attached to the hill surface left it

Turbulent diffusion from a point source in stratified and neutral flows around a thr~~imensioM1 hill-II 1977

1978 WiLLlAM H SNYDER and 3 f it BUNT

Yert 1c0l prof alas

Fig 8 Piume behavior in the absence of the hill; neutra1 Bow, high stack (F = CO,

H,/ft = 0.55)

at the point where the flow separated (generally

10&-l 10” from the upstream stagnation line, much as

happens to a plume in two-dimeesional flow round a

circular cylinder) Phrmes emitted close enough to the

stagnation i&s tended to be entrained into the wake

region and then rather rapidly regained their far-

upstream elevation while mixing through the depth of

the jump Beyond that point* these entrained phrmes

tended to be vigorously mixed ho~on~~y across the

wake, leading to the small wake concentrations evident

in the photographs Whether or not they were en-

trained, most plumes were affected by the vortex

shedding or low frequency oscillations of the wake

These wake vortices seemed to induce an oscillation in

the plume upwind of the hill, causing it to waft from

one side of the hill to the other This phenomenon will

be discussed in more detail later,

When the plumes were emitted at H, [Fig 11(b)],

they appeared to impinge directly on the hill surfact, as

opposed to rolling-up in an upwind vortex, and to

spread radially in ah directions The upper portions of

the plumes spread broadly in fingers to cover the entire top of the hill; the lower portions appeared to be rolled-up in the upwind vortex, hence, to be carried away from the immediate surface of the hill, then to be transported around the hill in spiralling patterns These plumes were much steadier in direction, &king the obvious wafting evidenced by those plumes re- fessed below H, The bulk of plume material appeared

to spread later&y from the impingement point and lose etevation continuously as it was transported round the hi& these plume segments were relatively coherent, remaining distinct for several hill heights downwind of the base of the hill and with little mixing across the wake

Plumes emitted above Ho, of course, were trans- ported over the hill top [Fig 11(c)] but if the release height was close to the dividing-streamline height, they spread broadly but thinly to cover the entire hit1 surface above W, Unlike plumes released at or below Ii,, plume material reached the hill surface only by diffusion perpendicular to the plume centerline Also,

Turbulent diffusion from a point source in stratified and neutral flows around a threedimensional hill II 1979

T;

02 -0.6

X

Vertccat profiles curves not Gausston

Pig 9 Plume behavior in the absence of the hill; neutral flow, low stack (F = a),

l Estimated using image source

the plume meander observed at or below Ho was (above H,) are obvious from the photographs (for compietely absent As H, was increased relative to If,, further discussion, see HS) These features, in combi- the point of first contact between the plume and the hill nation with the steadiness in plume direction, resulted surface moved toward the top of the hill; further in one of the largest observed surface con~nt~tions increases in H, moved the contact point to the lee side under any conditions Notice that, because of the lee-

of the hill The strong convergence of streamlines over wave+rduced flow separation on the lee side of the hill the hill top and close matching of streamline shapes (for F c l), we would not expect to find any concen- with the hill shape on the upper portion of the lee side tration of plume material below about H,

- Minimum

Y

(bi

F=U,/Nh

Fig 10 Plume characteristics in the absence of the hill for stack heights

H,/h > 0.39: (a) plume widths, (b) centerline concentrations at x = 0 (x

=

3.2.2 Source below dividing-streamline height

Concentration distributions measured on the hill

surface and with horizontal and vertical rakes in the

wake are shown in Figs 12 and 13, respectively The

height and lateral position of the rakes were adjusted

in an attempt to measure maximum concentrations

Figure 12(a) (F = 0.2) shows that maximum concen-

trations are found on the 180 or the - 165” lines,

whereas they are reduced by a factor of approximately

2 on the - 90” line and by 50 or so on the 0” line The

downward deflection of the plumes at the - 90” line

and the broader width of the plumes on the 0” line are

also evident

Figure 12(b) (F = 0.4) shows that a small offset of

the source in the lateral direction (y,/R, = 0.073)

markedly reduced the concentration on the 180” line,

but the maximum surface concentration was moved to

the - 90” line, with a value about 80 % of that observed

on the 180” line with no source offset

Figure 13(a) shows that lateral concentration distri-

butions in the wake are generally bimodal in character

as a result of the split plumes separating from each side

3.7 h)

of the hill and traveling nearly straight downstream thereafter An exception, of course, is the distribution from the offset source, where the plume clearly favors one side of the hill Concentrations measured with the rake are essentially the same as those measured on the 0” line, showing the vigorous mixing within the wake The vertical distributions [Fig 13(b)] show elevated plumes on the outside edge of the wake under very strong stratification (F = 0.2) but the elevation of the maximum concentrations are somewhat lower than the source heights (20-30 %) Plume elevations under somewhat weaker stratification (F = 0.4) are even lower ( < 0.5 If,) due to the larger downward deflec- tions (See HS, Section 4.3)

From many curves such as these, the peak value of the concentration I,,,~ = C,U,h2/Q as well as its location S,, on each radial line of ports was noted; they are tabulated in Table 2 Also, from such curves, it was possible, when the concentration dropped sharply on each side of the peak value, to define a plume width C, along each radial line These results are tabulated in Table 3 When H, > Ho, the plumes effectively travel-

Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill-II 1981

1982

-

I i I- L

Turbulent ditbion from a point source in strati&d and neutral flows around a three-dimensional Ml-II 1983

1984 WILLIAM H SNYDER and J C R HUNT

led over rather than round the hill, so that the

Open symbols, H,/h = 0.55; closed symbols, H&h

Open symbols, y,/R, = 0; closed symbols, y,/R, =

- 0.073

Fig 12 Concentration distributions measured on

the hill surface A, 180”; 0, - 165”;0, - 90”; 0, 0”

concentration on the 180” and 0” lines gradually

increased to a flat maximum before slowly decreasing,

much as it varies on flat terrain due to an elevated

source In thaf case, Zr does not have a useful physical

(b) Measurements made with verbcal rake

Fig 13 Concentration distributions in the wake of

the hill at x/h = 2.6

each radial line of ports at angle 0 is plotted for different values of offset y, (lateral displacement) of the source from the centerline for each Froude number (F

= 0.2 and 0.4) and each stack height (HJh = 0.39 and 0.55) As many as six runs were made under the same conditions; in the case of multiple runs, mean values of these ensembles of peak conentrations are plotted and the largest and smallest are denoted by scatter bars The salient points shown by these graphs are: (a) The maximum concentrations x, observed on the hill surface for 0 < y,/R, < 0.04 are between 40 and 104 y0 of the centerline concentration of the plume

Turbulent diffusion from a point source in stratified and neutral flows around a three~ime~sionai h&-11 1985

Table 2, Values and locations of maximum concentrations on hill surface and in wake of hiif

y,lR, z,,AL/~) x~~~/h) x~~s~f~~ xnu(&,x/fi~ ~~(Y~/~~

Horixontal rake*

XCL id@

*Rakes locatedat x/h = 264forall towing tank studies, F = 0.2,0.4,l.O;x/h = 194for wind tunnelstudies, F = eo.hand

y,/h are the value and lateral position of the maximum concentration For the elevation of the rake, see the adjacent column and note t

t xa and z/h are centerline concentration and elevation of horizontal rake

$ xW, y/k and z,/h are maximum concentration, lateral position, and location of maximum ~n~nt~t~on from vertical rake

at that hrterai position

8 R, in this case is zero The actual offset was - 3 cm

14.5(2.05) - X2.4(1.94) 1.03(2.05~

74.9(1.29) 7.14(1.46) 76.4(1.29) 3.12(1.62) 92.6(1.29) 5.89( 1.62) 17.1(1.24) 52.0(1.62) MO(O.81) 131(l.i4) UO(O.97) 130(1.19) 113(0.97) 134(1.24) 75.5(0.973 109(1.24)

29.lfl.19) 8.48(1.19) 20.7(1.41) 18.4(1.24) 36.0(O) 36.0(O) 20.7(0.65) - 0.92(0.32) 11.6(0.65) 0.95(O) 0.95(O)

lIG(1.41)

- lS(l.41)

2.04Q.14)

O&19) 4.33(1.08) 3.27(1.14) 0.98(1.08) 2.55(1.35) 2.03(1.14) 1.54(1.08f 150(0.22) 3.8(0.22)

0~~1.~3~

l.OS(1.29) 0.49( 1.49) 1.55(1.62) 1.98(1.35) 2.40(1.51) 2.12(1.35) 1.7611.19) 1.43(1.46) 37.1(1.03) 0.6(0.43}

0.9(1.28) 0.51(1.35) 45.5(0.32) 9.19(0.81) 3.09(0.65) 11.4(1.73)

-

514(O) 8.05(O) 1.03(0.43)

-

2.65( - 1.06) 2.31(-1.70)

3.71( 0.85) 3.57( - 1.28) 2.87( - 1.7)

3.95(- 1.70)

6.75(- 1.28)

- 3.83( - 1.06)

- 1.34( - 1.70) 6.05( - 1 C6) 6.58( - 1.06) 6.32( - 1.28) 7.10(-1.28) 5.15( - 1.28)

ll.o(-0.21) 8.11( -0.85)

G(O.13) 2.$- 1.5) (0) 1.12(0,13) 3.19( - 1.5)(0.25)

2.2&(0.30) 3.17( - 1.5)(0.25) 1.87(0.30) 3.43( - 1.7) (0.30) 0.79(0.30~ 2.87( - 1.7) (0.30)

1.36(0.30) 395( - 1.7) (0.30)

2<2(0.30) 6.6( -f.S)(O.l7) G6(0.44) 3.83( - 1.1) (0.44) 1.07(O) (0.44)

0.89(0.~) 1.18( - 1.5)(0.40) 3.01(0*43) 3X(-1.7)(0.47) 3.4qO.43) 3.09( - 1.7) (0.47) 1.98(0.44} 6.40( - 1.5) (0.49) 2.84(0.43) 3.74f - 1.5) (0.45) 1.99(0&q 5.07( - 1.5) (0.44)

1 5.24( - 1.5) (0.31) 10.9(1‘0) 192( -0.64) (0.85) 3.32(0.81) 8.99( -0.64) (0.94)

l*& 1.91) 1.45( - 1.06) 2.26( - 1.91) 4.13( - 1.70) 1.78( -0.85) 1.35(-0.85) 4.28( - 1.28) 5.21(-128) 5.07( - 1.06) 5.43( -0.43) 7.09( - 8.5)

3*33(-l.%) 6.75( - 1.28) 1.73(O) 1.51(-0.21) 1.70( -0.21)

- 4.72( -0.21)

1.83( -0.43) 2.30( -0.26) 2.75(O)

-

-(0.13) 0‘88(0.303

- (0.21) 0.%(0.21) 1.34(0.30) 1.03(0.44) 1.54(0.30) 1.38(0.30) 0.3(0.30) 3.21(0.77) 1.21(0.77)

0.9:-I‘S) (0) 0.74( - 1.5) (0.09) 278( - 1.7) (0.17) 4.13(-1.7)(0.21) 6.30( - 1.5) (0.09) 0.28( - 1.5)(0.31) 4.76( - 1.3) (0.26) 5.80( - 1.3) (0.26) 5.07( - 1.1) (0.30) 4.%( -0.6) (0.77) 7.18( -0.6) (0.64)

0.23(0.13) 1.41(-1.5)(O) 0.38(0.13) 6.47( - 1.5) (0) 1.73(0.30) 0.66( - lS)(O.O9) 0.29(03Of 0.37( - 1.5) (0.26) 1.25(0.30) 0.5( - 1.5)(0.17)

-

1.67(0.43) 1*88(O) (0) 2.15f0.43) 2.19(O) (0.21) 2.75(0.43) 8.2(O) (0.81)