Turbulent diffusion from a point source, laboratory data

Turbulent diffusion from a point source, laboratory data

~mcupkric EnviroMvnr Pnnted in Great Britain Vol IS No IO, pp 1969-2002 ocKJ4-6981/84 13.00 + 0.00 Pergamcn Press Ltd 1984 TURBULENT DIFFUSION FROM A POINT SOURCE IN STRATIFIED AND NEUTRAL FLOWS AROUND A THREEDIMENSIONAL HILL-II LABORATORY MEASUREMENTS OF SURFACE CONCENTRATIONS WILLIAMH SNYDER* 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 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 A B c r 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) Brunt-VPisiihi frequency emission rate from source s sh cmx C” d D F ff HD I L n “s S AY c e P =n *On assignment from the National Oceanic Atmospheric Administration, U.S Department Commerce t Also Department of Engineering and of u; 1969 radial coordinate (origin at center of hill) radius of hill at height of source along-plume coordinate measured from source (see Part I) 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 distribution in absence of hill standard deviation of vertical concentration distribution in absence of hill WILLIAM H SNYDER plume width (standard deviation) in azimuthal direction along-surface plume width defined as distance between points whereconcentration 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 in absence of hill nondimensional concentration, Ct’,P/Q plume centerline (nondimensional) concentration in presence of hill maximum (nondimensional) surface concentration in presence of hill maximum nondimensjonai plume centerline concentration in absence of hilf maximum nondimensional plume centerline concentration in absence of hill assuming a best-fit Gaussian approximation to the measured distribution nondimensional concentration in wake of hill INTRODUCTION The structure of strongly stratified flow over a threedimensional hill may be usefully envisioned as composed 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 demonstrated through laboratory experiments by Riley et al (1976) Brighton (1978) and Hunt and Snyder (1980) (hereafter referred to as HS) HS performed sah-waterstratified towing-tank studies using a simple belfshaped 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 H,jh=l F, (1) 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 h (h - 2) (- dp/dz)dz, (2) f&(H,)=g “n and J.C R HUNT 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 discussed above using the boundary conditions applicable 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 steadystate 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 stratification (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 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 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 concentrations were estimated by assuming constant diffusivities, while Bass rt ul (1981) used Gaussian plume assumptions with the Pasquilf-Gifford curves (Turner, 1970) Further, since any streamline found from potential 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 Simplification of plume impingement on threedimensional stratification trations on the surface of a series of 20 bluffs whose maximum slopes range from 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 straightforward 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 dispiacements 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 threedimensional hill in neutral and stably stratified flow and to compare these with those on level ground and on two-dimensional hills hill under very stable 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 conducted 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 turbulence 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 conditions, the diffusion of chimney plumes is largely determined by initial mixing within the plume (Weil, 1983), the present experiments are a reasonable simulation of such conditions In neutral conditions, our experiment is not a good simulation of the full-scale adiabatic boundary layer, but we include results obtained 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 dflusion in neutrally stratified turbulent flows over hills of WILLIAM 1972 H SNYDER and 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) 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, 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 = 180, - 16.5,- 90 and 0” (see Fig 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 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 Test I tubes J C R HUN.T 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 establish 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 9,, it permitted the measurement ofdilutions between 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 standards 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, 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 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 background 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 for collecting dye Efftuent reservoir / Verticoi and twirontol mke Fig Schematic diagram showing system for collecting dye samples in towing tank Turbulent diffusion from a point c E t - Modified Beer’s low fit A 001 source in stratified and neutral flows around a three-dimensional hill II \ I Standards s t o.oQJ’ 0.04 CIO2 ’ ’ 006 ’ ’ ’ ’ 0.08 0.10 0.12 014 ’ 016 11 0.16 0.20 Calorimeter outtut Iwltsl Fig Calibration of calorimeter 2.2 Concentration measurementsin 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 detector) 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 ppm total HCs) and a 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 connecting the tubing from the HC analyzer sequentially to each of the surface ports (see Fig 4) PRESENTATION AND DRTCURRION RESULTS OF 3.1 Concentration distributions hill in the absence of the Because it was of interest to compare concentrations measured on the hill surface with those on the 1973 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 circumstances in the absence of the hill The baseline concentrations 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 shows typical behavior of plumes emitted above the surface boundary layer under strong stratification (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 turbulence and creating a gravity current (see later discussion) 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 significantly 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 Concentration measurements in wind tunnel (dimensions in cm) 1974 WILLIAM H SNYDER and J C R HUNI l - - ~~ ~ Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill II width, however, was small but positive Figure 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 horizontal 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 and show the concentration distributions 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, showing 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 and Fig 10 The table is divided into two parts: (a) H,/h = 0.13 and (b) H,/h L 0.39, because the highlevel plumes were quite similar to each other, yet quite different from the low-level plumes Visual observations, 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 = 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 turbulent), 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 = 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 Z; = d(3~/4 +~(3n/4)“~/hF)‘!~ (4) 1975 Hence,XC,” F-“’ andZ,,“a F”2whenx/hF% a l.For our conditions at F = 0.2, these estimates yield Z,“/h = 0.15 and Xi/h = 0.0046 The observed value of X:y” is about twice this estimate, due to the internal turbulence of the efBuent as it exits from the stack and also the ambient turbulence created by the flow around the stack The observed value of Zz is about 10 times that predicted by the gravity-current estimate but does have approximately the F “’ behavior In fact, if it were not for the turbulence (the second mechanism of spreading), no dilution would occur, so that XL would equal its value inside the stack, namely 1725 The turbulence has produced a dilution of 1725/243 = 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 < 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 horizontal 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 elevation as they were transported around the sides of the hill Plumes that were attached to the hill surface left it _-_ I-z T; 0.0, 0.1 - 0.2 - 03a profiles Vertical I 0.1 profite X Curve not Gaussian Lateral x at x =0 at x =0 Fig Plume behavior in the absence of the hill; moderately strong stratification, stack at top of boundary iayer (F = 0.4, H,/h = O.t3) Side view Top view Turbulent diffusion from a point source in stratified and neutral flows around a thr~~imensioM1 hill-II C L E >C d 1977 1978 WiLLlAM H SNYDER f it BUNT and Yert 1c0l prof alas Fig Piume behavior in the absence of the hill; neutra1Bow,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 farupstream 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 entrained, 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 refessed 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 transported 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, 1988 H SNUYDEH J c‘ K HCN and WILLIAM deflected downward by an amount at the separation point of A = 4F2h (see Equation 2.3 and Table of HS) As the plume is entrained into the wake and is advected downstream of the hill, it rises back to near its original height H, Hence, it is mixed through a height of about A, and we may estimate its thickness as C, Ez + A 4Ef + A For F = 0.2 ZFjh = 0.05 (Table l), the theoretical value of A/h is 0.08, and the observed value is about 0.2, so that the predicted values of X,,/h are 0.28 and 0.4, using theoretical and observed values of A/h, respectively These predicted values are quite close to those observed (cf., Table 3) The data of Fig 14 are replotted in Fig 15 as the variation of Xmx = 180’ as a function of (y,/R,)‘, on since a first hypothesis is that x varies with y, in much the same way that x at x = 0, J‘ = 0, would vary with the lateral displacement of the source ys in the absence of the hill, i.e x(Y,) = x(0) exp (- r$W where is the plume width (standard deviation, cY) While the data show considerable scatter, the use of the plume width in the absence of the hill (a!) clearly does not fit the data Concentrations on the 180” line decreased very rapidly as the source was moved laterally from the centerplane, much more rapidly, in fact, than could be accounted for by our hypothesis A better fit to the data suggests a much narrower plume, i.e with width only half the value measured in the absence of the hill The theory indicates that I z 0.80; (cf., Figs and of Part I) This finding is contrary to the measurements using a line source upwind of a circular cylinder in large-scale wind-tunnel turbulence by Britter and Hunt (1984) who found that 12 6; Perhaps a more practically useful investigation is to study the effect of source offset on the maximum F H,/h a02 002 004 004 039 _ also plotted the theoretical curve derived from potential flow theory (cf., Fig 4, Part I) as well as the concentration expected on the 180’ line in the absence of the hill The decrease with _rr is similar to that predicted by the theory of Part I, if all values of x are reduced by an empirical factor of about We believe the main reason why all the observed values of x, are less than the predicted values is because the centerline of the plume slowly fluctuated either side of the stagnation line of the hill This phenomenon was observed by Riley et al (1976) who found that the period of the fluctuations was about equal to the period of vortex shedding In Fig 17(a), photographs are presented of the top view of the plume at a number of stages in a typical oscillation The first of these may be compared with the theoretically derived plume outlines in Fig of Part I In Fig 17(b), the lateral displacement of the point of impingement of the plume from the centerline AL’ is plotted as a function of time Observe the highly irregular oscillation; clearly there is a significant component of energy of oscillation at a frequency corresponding to the vortex shedding period of 33 s, calculated for the hill radius at the stack height But since this period differs with elevation and since the flows at each level in the wake are not disconnected, some overall, apparently irregular, oscillation occurs At very low Froude number (F < 0.1) (and lower Reynolds number) Brighton (1978) found that regular and coherent vortices were shed with one wavelength; our wake vortices were observed to be much more irregular The effect of the plume’s oscillations on the mean concentration can be estimated from the ob- 055 - xrnX the hill Since no stgnttion cant effect of Froude number between 0.2 and 0.4 on x mx was observed in Fig 15, we plot I,,,~ (wherever It occurred) as a function of y,/R, for the two different stack heights H, in Fig 16 For comparison, we have 039 _ surface concentration - 055 X/X”.= exp L-y: 121’1 -(,;/R,I’ j 001 I 0001 I 0002 I 003 I 0004 0005 ( y, / R5 I2 Fig 15 Maximum surface concentration on stagnation line (180”) as a function of source offset )6 Fig 16 Maximum surface concentration on hill as a function of source offset H,/h = : A, 0.39; El, 0.55 Potential flow theory, Part I, with xI = - 3.88 R,and D = 0.0005 Concentration on stagnation line, but in absence of hill Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill II (0) Sequential time photographs F=O.2, Fig 17 (a) H,/h=O.39:y,/R,=0.032 1989 1990 WILLIAM H SNIDER and J C R HC;NI of order RJU, could produce a 50”, reduction mean concentration at the stagnation point III the 3.2.3 Source above dividing-streamline height; F < 0’ a* \ cl ci -0, -32 -03 / cl 50 100 150 Time 200 250 300 Csecl (b) Displacement as a function of time P = 0.2, H,/h = 0.39, y,lR, = : 0, 0.032 @y/R, = 0.53, o = 0.14);0, 0.064 @F/R, = 0.13, (r = 0.077) Fig 17 Oscillation of plume impingement point served amplitude of the oscillations, the shape of the plume near the stagnation point, and the mean concentration profile (see Appendix) It is not certain that these wake induced fluctuations would be found in the atmosphere, but it is quite likely that low frequency fluctuations in wind direction would occur An analysis of our observations (see Appendix) suggests that a random change in wind direction of as little as 0.1 radians (6”) over a time scale 3.2.3.1 Stack height equal to hill height; variable Froude number The maximum normalized surface concentration when H, > H, (F = 0.2, H, = h) was found to be 150, i.e 62 “i, of x”, the concentration in the absence of the hill This maximum occurred on H = 0” at a distance of 0.22 h along the surface downstream of the hill center The distribution along the hill centerline (Fig 18) shows a very limited contact area The explanation for the large concentration and limited contact area is evident from study of the plume photographs in Fig 19(a) The dividing-streamline height here was H, = 0.8 h, so that the plume was released (H, = h) not far above H, The strong convergence in the vertical direction and divergence in the horizontal as well as the limited downward penetration of the plume on the lee side of the hill are apparent Because the flow is highly divergent in the horizontal plane, the streamlines remain very close to the surface and therefore, as the three-dimensional neutral flow analysis of Part I indicates, the concentration can be of the same order as the undisturbed value on the plume centerline In many respects, this plume behaves as if it were released from x/h Fig 18 Surface concentration profiles with HJh = H&h = : La 0.8; 0.6 Hill shape shown in lower part Turbulent diffusion from a point source in stratified and neutral flowsaround a three-dimensional a stack of height H,-H, upwind of a hill of height h -H, = Fh, as if a ground plane were inserted at i.e height H,.Thestratification above H,,of course, has important influences on the diffusion as well as the vertical convergence and horizontal divergence of the streamlines, so that the 5ow structure over the hill of height Fh must be treated like that with F = 1, i.e the Froude number of the 5ow over this smaller hill is unity, (Bass et al., 1981; Hunt et al., 1984) Plume material in this case reached the hill surface not by direct impaction, but by diffusion normal to the plume axis Plume meander as was observed for cases with H, H,) was also noted in field observationsat a 105 m hill @very et al., hill II 1991 1982), but may also have been associated with pronounced low frequency oscillations in the wind direction below 1CKlm As the Froude number was increased to 0.4, this plume rose higher above the hill top and therefore made first contact with the hill surface slightly farther downstream; also, it was swept farther down the lee side before leaving the surf&e as the flow separated [see Fig 19(b)] at an elevation very close to H,(= 0.6 The concentration profiles of Fig 18 h) demonstrate these features quantitatively and also show that the maximum surface concentration was somewhat reduced from the F = 0.2 case, i.e x,/x” = 0.25 At F = 1.0, this plume cleared the hill by a wide margin with no surface concentrations [Fig 19(c)] In , (a) F = 0.2 (H&I = 0.8) Fig 19 (a) _ _ _ _,,/ I 1992 WILLIAM H SNYDER and J C R HUNT _ (b) F = 0.4 (H&I = 0.6) Fig 19 (b) the top view, there is a suggestion of lateral plume divergence over the hill top and slight convergence as it is swept over the lee slope 3.2.3.2 F = 1; variable stack height Table shows that when F = and H, = 0.13 h, the maximum surface concentration was 30, an inorease of 35 % over its value when F = 0.4, the location of this maximum being at a height of about 0.4 h on the upwind centerline In contrast, the maximum concentration when H, = 0.39 h was 44, a reduction to about 40% of its value when F = 0.4 The location of the maximum was on the top lee side of the hill, because the plume just grazed the top Hence, it is not surprising that when H, = 0.55 h, the plume passed over the top and the maximum concentration was much smaller, xmx= 11 (7% of its maximum value when F = 0.4) The position of the maximum was on the lee side at a height of about 0.13 h As discussed earlier, when H, = h, no surface concentrations were detected, although the wake concentrations measured by the rake showed values similar to those at small Froude number (x, Z 5) Figure 20 shows the concentration distributions on the hill centerline for the above stack heights (it also includes neutral flow, wind-tunnel results to be discussed later) It would appear that the maximum surface concentration would occur for the lowest stack height that is above the surface boundary layer; plumes Turbulent diffusion from a point source in stratified and neutral flows around a thr~-dimensional bill-41 (c) F = (if,/h I993 = 0) Fig f Variation of plume behavior with Froude number; HJh = I O from lower stacks would be dispersed and diluted in the surface boundary layer before reaching the hill slope, and those from higher stacks, of course, diffuse through larger distances normal to the plume axis before reaching the hill surface Also, the area of coverage of plume material on the hill surface is dramatically reduced as the stack height is increased (cf., Fig 11, F = 1, H,/h = 0.5 and 0.8) 3.2.4 Neurrai Pow The results of the measurements of concentration distributions downwind of point sources in the wind tunnel are shown in Figs 8,9 and 21 The location of the stack was the same as in the towing tank, 3.7h upwind of the center of polynomial hill The wind tunnel data are presented in lieu of the neutral towing tank data because, due to the relative ease ofwind tunnel measurements, thesedata are much more extensive and possess better accuracy and resolution In Figs and 9, vertical concentration profiles were shown downwind of the source when its height H,/h was Cl.13 and 0.55 Note that the boundary-layer thickness was 0.28 h at the location of the hill, so that only the lower plume was diffused by boundary layer turbulence, the higher plume being diffused by turbulence generated in the wake of the source tube The curves shown are Gaussian distributions that ‘best-fit’ WILLIAMH SNYDER and J C R HUNT 1994 ; o.;/{ -2 -I I x/h Fig 20 Hill centerline concentrations Open symbols, neutral flow; closed symbols, F = HJh = : A, 0.13; 0, 0.39; 0, 0.55 the experimental data, i.e they have the same standard deviation as the experimental data set and the maximum concentration xmxl was adjusted for best-fit In Fig 9, the curves are ‘best-fit, reflected’ Gaussian curves, which means that both uz and x&s were adjusted to provide a best-fit to the experimental data (assuming the plume centerline height was equal to the release height) The parameter I&, is listed in Table The maximum ground-level concentration xmr for H,/h 0.13 was found to be 15 (when normalized with = respect to the free-stream speed and the hill height) This compares well with the value of 14, based on the conventional Gaussian prediction (Pasquill, 1974) measurements and then compare them with the theoretical predictions of Part I For the lowest stack height, H,/h 0.13, value of = the on the surface was 23, an increase of SOT/, XmX compared with level ground Whereas the position of the maximurl was located about 30 stack heights downwind of the source with no hill, it was only about 20 stack heights downwind with the hill and its location on the hill was about stack heights above the base A remarkable feature of this plume can be seen from the surface concentration contours: it effectively splits in two, passing either side of the hill As the stack height was increased to 0.39h, the maximum surface concentration remained about the same (x, = 21.2), but its location moved to the hill C = 2(az/u;)QI[enUH,7, (6) top This concentration is about one-third of the plume and the experimental observation that ui = try” centerline concentration and 14 times the maximum value one would estimate for a stack of this height on (Table 1) level ground on the basis of the usual formula (6) We The ground concentration below the upper plume also note that (1) the plume passed over the top of the did not reach a maximum value it is relevant for hill, (2) the plume was thinner at the hill top than it was comparison with the flow with the hill to note that at upstream (a,/h z 0.03 compared with 0.04), due to the hill center (x = 0) u~/!I 10.05; thus Q, was only convergence of the streamlines and (3) the plume is about 1.8 times the stack diameter also wider than upstream by roughly 30% when The vertical, horizontal and surface profiles of mean = concentration near and on the hill are shown in Fig 21 x/h -2, due to the horizontal divergence of the for four source heights H,/h 0.13,0.39,0.55 and 1.0 streamlines = A remarkable reduction in 2, (from 21.2 to LO),was We first indicate the main qualitative features of the Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill II I t i; _ -iS x/h (Of H,/h=O.t3 I fop view I -3’6 (b) H./h=039 Fig 21 (a)-(b) 1995 1996 WILLIAM H SNYDER and J C R HUNT I iY - _ [C) H,/h=O Y - h 07 z/h=098 *+. ._ :_L.-A-‘_’ - t 55 - ‘/ c x/h Id) H,/h=l Fig 21 Composite view of concentration field over polynomial hill in neutral flow; vertical and lateral profiles and surface isoconcentration pattern Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill II observed when H,/h increased from 0.39 to 0.55, a factor of 1.4 On level ground the concentration would be expected to decrease by a factor of about for such an increase in source height A qualitative explanation is provided by the loci of the positions of maximum concentration in a series of vertical profiles, as shown in Fig 22 The distance from the hill to this locus ns increased by a factor of 2.5, from about 0.043h about to 0.11h, as H,/h increased from 0.39 to 0.55 No measurable surface concentrations were observed when the stack height was increased to l.Oh An important feature to note from the concentration profiles for all source heights (except H, = h) is that there is a wake behind the hill across which the concentration is roughly constant in the lateral and vertical directions This wake region, termed an ‘envelope’ by Huber et al (1976), is a region of vigorous large-scale turbulent mixing; it is deeper and more extensive than the region of reverse flow shown in Fig 15(c) of HS An interesting result is that x in the wake 4h downwind is about 1.0 for the lowest stack heights This is unlikely to be a universal result, as it certainly differs from the case where line sources are placed upwind of a cylindrical body in a turbulent Bow (Puttock, 1979) The stack with height H,/h = 1.0 was too high for its plume to diffuse downwards into the wake, as is quite clear in Fig 21(d) Figure 20 provides a useful comparison of surface concentration profiles under neutral (wind tunnel) and stratified (F = 1) flow conditions These concentrations have been normalized by the maximum concentrations found at the center of the plumes in the absence of the hill (x = 3.7h) Several interesting features should be noted (1) The point of flow separation on the lee side of the hill under neutral conditions is quite evident; a sharp decrease in concentration was observed at x/h 0.5 (z/h Z 0.9) for stack heights of 0.13h and 0.39h By contrast, at F = 1, the flow swept the plumes from the lowest three stacks continuously down the lee side of the hill because it did not separate from the surface, so that the changes in concentration along the lee surface were relatively smooth For H, = 0.55h, the surface concentration profile became flat beyond the separation point Evidently, the plume from the H, = h stack was not entrained into the recirculation region, because no surface concentrations were observed with that stack (2) Upwind of the separation point, the effect of the stratification appears to be negligible; the concen- 1997 trations generally differed by much less than a factor of Note that the absolute concentrations differed considerably because of the differing diffusivities in the wind tunnel and in the towing tank (i.e Table l), but this difference has been largely offset by the normalization by XL The maximum surface concentrations occurred upwind of the separation point for the two lowest stacks, and the value and location of the maxima appear to be essentially independent of the stratification (3) Downwind of the separation point in neutral flow, the neutral-flow concentrations are constant, indicating strong turbulence and mixing in the recirculating region As mentioned above, in the stratified flow, the surface concentrations continuously (and smoothly) changed because the flow did not separate For the two lowest stacks, the maximum occurred upwind of the (neutral flow) separation point so that the concentrations decreased with distance downstream For H, = 0.55h, however, the concentration increased with downwind distance, reaching a maximum near the downwind base of the hill with a value 10 times the maximum observed with the same stack height under neutral conditions (4) The decrease in maximum surface concentration was only a factor when the stack height was increased by a factor of (from 0.13h to 0.39h) Theoretical arguments suggest that, over level ground, this decrease would have been Hi/H:, or a factor of How these results compare with our theoretical ideas of Part I, and the ideas put forward by Egan (1975) and Egan and Bass (1976)? These theoretical suggestions were strictly applicable only lo unidirectional flows over axisymmetric obstacles, but were assumed to have some validity for non-axisymmetric three-dimensional hills The following expression was derived in Part I for the ratio of the maximum surface concentration C, to the plume centerline concentration C” in the absence of the hill - W,lh)’ ’ exp 8(a,/h)2(ti(s,)/U,)Z ’ where s,, is the distance along the streamline from the source to the hill top, Z_ the half-length of the hill, and is c(s,,) is the velocity at the hill top (Note that Equation (7) differs from Equation (24) of Part I because in Part x/h Fig 22 Loci of maximum concentrations in neutral flow H,/h = : A, 0.13; QO.39; o, 0.55; 1.0 1998 WILLIAM H SNYDER and I (2s,/L)/(~(s,)U,) was taken to be of the order of and because there was a typographical error in the exponential function) In the following discussion, we will examine in detail the assumptions that were made in deriving this expression The first assumption was that the vertical profile was Gaussian with an ‘image’ profile centered an equal distance below the hill surface Thus, if u,, is the observed plume width (normal to the plume axis), xc the center-line concentration, nn the distance of the plume centerline above the hill surface (also normal to the plume axis), then on the surface we should expect = 21~exp [ - t$2bi] (8) For the neutral flow cases with H,jh = 0.39 and 0.55, this formula provides reasonable estimates for surface concentrations using the measured values of nn, cr, and xc at the hill top (see Table 4), but the values predicted are highly sensitive to the ratio n,/a, For example, when n&, is 1.5, an overestimate of nI by lo‘& combined with an underestimate of 0, by 10% results in an underestimate of surface concentration by 43 7; Hence, the relatively good agreement between predicted and observed surface concentrations shown in Table is probably fortuitous The value of u, required to predict the observed surface concentration is also shown in the table, and it differs from the measured value by less than mm, which is beyond the resolution ability of the instrument carriage in the wind tunnel The second assumption in our theoretical estimates was that the turbulence (or the eddy diffusivity) was unchanged by the hill, but that the plume depth and width were decreased and increased, respectively, by the vertical convergence and lateral divergence of the streamlines (see Fig of Part I) From the observed mean streamline pattern (Fig 11of HS), we see that the vertical distance between streamlines is about halved as the flow passes over the hill for streamlines starting at HJh ‘c 0.5 This is not as great a reduction as predicted by potential flow theory for flow over an axisymmetric body, where the distance between streamlines (and the plume width) is one third of its upstream (or undisturbed) value However, the observations of the plumes when H&h = 0.39 and 0.55 suggest that a,/h is about O.M.6 times its undisturbed value (cf., Tables and 1).This may be partly due to the special nature of the stack’s wake turbulence diffusing the plumes and due to the fact that, as the plumes are brought nearer the hill surface, they enter the surface boundary layer and are slightly thickened Also the low-frequency wake eddies wave the plumes around somewhat, broadening them Thus we conclude that the theory is correct in predicting an appreciable reduction in plume depth, but may overestimate the reduction The third assumption necessary for our speculative estimate for the effect of source height on maximum surface concentration (7) is an estimate of nJ, the distance from the hill to the center of the plume J C R HUKT Taking the flow round a sphere of radius h as a rough guide, then n,/h Z (H,/h)zs3; for H,/h = 0.39 the estimated value of n&h is 0.05, the measured value being 0.04; for H,/h = 0.55, the estimated and predicted values are both 0.10 For H,/h = 1.0, the estimated and predicted values are 0.33! The fourth assumption is that the ratio of the azimuthal plume width over the hill to its value in the absence of the hill (a;/$) is approximately equal to V&/h)- ‘7 at least for hills axisymmetric about the xaxis The observed ratios of a;& for the various sources and the theoretical estimates for potential flow around a hemisphere are shown in Table The measurements confirm the usefulness of the simple eddy diffusivity theory for estimating ui/u: and the estimate (H,/h)- ’ based on purely krnematic arguments Thus the general assumptions of our Part and Egan and Bass’ papers for surface concentrations on three-dimensional hills in neutral conditions are reasonable That is not to say that the actual predictions are particularly accurate, or are even likely to be particularly good in the most sensitive range of the ratio HJh Assuming potential flow to estimate nr, (so fi(s,)/CJ, = 1.5) and taking the observed value of (u,/h) z 0.03, the order-of-magnitude theoretical estimate (7) suggests that the ratio of ,Y,, when H&h = 0.39 to its value when HJh = 0.545 is (0.39JO.545 x exp [( - 0.39)4/(8 x 0.03’ x 1.5’)]/exp [ - (0.545j4/ (8 x 0.03’ x 1.5’)] = 40 An alteration of (u,,h) by +0.002 ( + 0.5 mm) can change this ratlo from 24 to 62 The observed ratio is about 20 Thus, this theoretical prediction gives a useful estimate for the extraordinary sensitivity of maximum surface concentration to the height of a source near a threedimensional hill This equation also suggests the ratio of the surface concentration to the plume centerline Table Comparison of predicted and observed hill surface concentrations at x = (Hill center) -_I_ H,lh Observed n,/h Observed u,/h Observed xc Observed ,Y_ Predicted x on/h required to yield observed x 0.545 0.10 0.031 48 0.39 0.37 0.03 I 0.39 0.038 0.028 30 17 23 0.025 Table Ratios of horizontal plume widths with and without the hill _ H,lh Observed +; Theory Estimate (H,/h)- ’ (strictly for H,/h $ I) 0.13 0.39 1.0 6.0’ 3.1 1.2t 6.9 2.31 1.23 7.7 2.6 1.0 * From surface concentration over hill t From concentrations on center of plume Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill II value in the absence of the hill At H,/h = 0.39, the theoretical estimate is 0.21, compared with the observed ratio of 0.3 It is perhaps of interest to use our simple eddy diffusivity computations for the flow over a hemisphere to estimate actual concentrations for the polynomial hill Choosing the diffusivity D = 0.0005 so that (an/h) = 0.0275, then x(x = 0) = 9.6 (cf., 16) when H,/h = 0.39; x(x = 0) = 0.9 (cf., 0.4) when H,/h = 0.55; and x(x = 0) = 10m9 (cf., < 10m4), when H,/h = 1.0 The rough agreement with the measurements suggests these simple models may have some predictive value if crude estimates are required The variable diffusivity model gives similar answers because the mean flow straining dominates the diffusion process 3.2.5 Further discussion Figure 23 presents an overview of the maximum surface concentrations measured in H,/h x F space Overlaid on this graph are the dividing-streamline height (H,/h = -F), the boundary layer, and somewhat speculative isoconcentration lines These isoconcentration lines were drawn to fit the data as measured, but a generous allowance was made for scatter in order that they might also satisfy our physical intuition The graph suggests that the largest concentrations occur when the source is released near the dividing-streamline line height and that they decrease rapidly with distance to the right of this line (larger stack heights or Froude numbers) This rapid decrease is due to the fact that, as the stack height or Froude number is increased, the contact point moves upward over the hill top, then down the lee side; further increases in H, - H,, much above the thickness Zz of the plume over the hill result in the plume lifting off the surface, with no contact at all Note that if H, -H, ~1Z,” a significant surface concentration can arise because of the downward deflection of the !!t h 06 Fig 23 Isoconcentration contours (dashed lines) in Solid line represents H,/h = -F Numbers represent concentrations measured at those points Stippled area represents surface boundary layer Striped area represents roughly uniform concenl tration t indicates Z,“/h, the depth of the plume at the location of the hill H,/h x F space 1999 streamlines onto the hill If the source height is less than the dividing-streamline height (to the left of the line), the measurements suggest that the maximum surface concentrations are roughly uniform in this region But visual observations during the tows suggested that (a) the lower the stack height, the more prominent was the plume meander (hence, lower concentrations), but (b) the actual surface concentration seemed to depend upon the precise point at which the plume entered the vortex on the upwind slope The size and elevation of these vortices appeared to vary from one tow to the next (indeed, even during a single tow), and were not investigated in detail Fcr the very small stack heights, the plumes dispersed relatively quickly in the surface boundary layer, so that concentrations were essentially independent of the Froude number This Froude number is a bulk quantity that characterizes the overall flow over the hill; the Froude number characterizing the boundary layer flow is much larger both because of the smaller scale of the boundary layer (6 h) and the reduced density gradient caused by the mixing CONCLUSIONS (1) To what extent can these experiments be used to predict the dispersion of pollutants from sources near hills in the atmosphere? When the atmosphere is very stable, photographs ofchimney plumes suggest that on many occasions the broadening of the plume is determined largly by the plume’s own turbulence and, to a lesser extent, by ambient turbulence (Weil, 1983) That is the type of diffusion modeled in these experiments The most stable case we examined (F = 0.2), where Xy/R, = 0.3, X,/R, z 0.05 and x,/R, 2: - 3.5, would correspond to a chimney located 3.5 km (x,) upwind of a hill with a radius at the stack height (R,) of about km, the plume being about 50 m deep (E,) and 300m wide (Z,) In fact, such plumes in very stable conditions, diffusing due to their own turbulence, are more likely to have a value of XLof about 30m The plume dimensions in our experiments more closely correspond to a plume being spread by weak atmospheric turbulence (F stability), when Zz would be about 70m and Z,, would be 400m (2) The broadest conclusion that can be drawn is that when the source is below the dividing-streamline height, the plumes impact on the front surface, yielding surface concentrations x,/x” Z 1, the largest value being about 1.1 However, low frequency eddying in the wake can affect plumes upwind of the hill so as to increase their area of impingement on the hill and lower the average concentration A simple model provides a quantitative estimate for this effect once the actual amplitude of plume oscillations is known The eddying effect appears to become more significant as the source height is decreased (relative to the dividingstreamline height), but additional experiments are needed to confirm this 2ooo WILLIAM H SNYDER and J C R HC’NJ (3) Although no concentration measurements were made for a case with H, = H, the plumes were observed to impinge quite steadily on the surface, to spread radially in all directions from the impingement point and to cover the entire hill surface above Ho with a thin layer of dye Because of the much reduced plume meander (compared with plumes from lower sources) we would expect surface con~ntrations x,,/x z 1, or even slightly larger (4) When the source is above the dividingstreamline height, the plume surmounts the hill, but if it is only slightly above Ho, the plume strongly diverges in the direction parallel to the surface, and diffuses quickly to the surface with, again, x,/x’ z As the source height is increased the location of the maximum surface concentration moves upward on the upwind slope to the hill top, then down the lee side Further increases in H, result in rapid decreases in x,,/x” But there may be values of HJh, in this case, HJh ~0.4, where even when F ) 03, X,/X” l/4 The basic reason is that when F -+ cc, the distance from the surface of a roughly symmetric threedimensional hill of streamlines through the source is a factor of about 1/3(H,/h) of the distance when the source is over level ground (when H,/h < 1).We would expect xmx/xc to decrease gradually as the slope of the hill is decreased On the other hand, if the hill has a stagnation point on its upwind slope, as there would be on a steep three-dimensional hill, then a source on the stagnation streamline might produce a surface concentration x, z xc, in much the same way as a plume impinging on a building (Hunt et al., 1979) (5) Theoretical models for the flbw and diffusion processes at low and high Froude numbers support the above generalizations (2 and 4) Even so, more experiments are needed to confirm them, and the theory particularly needs extension to cases where H, = H, and where hills have small slopes (6) As mentions above, when the source height is below the dividing-str~mline height (H, < H,), the location of the maximum concentration is on the upstream face A small displacement of the source off the stagnation streamline (by less than c;) does not appreciably change the magnitude of K,, but can move the location of x,x to the side of the hill Consequently, slow (i.e on a time scale > x,/U) and small oscillations in the wind direction can cover the hill with the maximum concentration These oscillations, whether caused by vortex shedding in the wake or wind meander, can signifi~ntly reduce the average value of xrnx When Ho < H, < h (F c l), the maximum concentration is found on the top or the leeward side of the hill (7) When H, -c H,, the concentration in the recirculating wake region x, is approximately constant and is likely to be much smaller than xmr due to enhanced vertical mixing ( - 4F’h) and much enhanced horizontal mixing ( * 3R,) in the wake (In our experiments, x,/x, was typically I/SO.) Consequently, downwind of the r~irculating region, a reasonable (The factor of l/2 is included toaccount for an ‘average wake velocity’ of half the far-upwind speed) This estimate was also relatively successful in predicting the surface con~ntrations on the lee side of the hill; but, note that the area of coverage is limited to a band extending vertically over a depth of about 4F2h and horizontally from the separation point on one side of the hill to that on the opposite side (i.e about &-80’ from the downwind centerline) At high Froude numbers, a three-dimensional hill produces a clearly defined ‘envelope’ within which the concentration is appreciably constant (8) How these results compare with the current EPA Valley model (Burt and Slater, 1977)? At low Froude number the high surface con~ntration and the form of the impinging plume support the general assumption of the Valley model, although in quantitative terms our measurements were about l/2 of those predicted by the Valley model In the absence of plume meander, the concentration would have been, we estimate, about 2/‘3 of the valley model prediction Consequently we concluded that, at low Froude numbers, the Valley model gives a conservative estimate, at least for short-term averages (e.g f h) The Valley model includes factors for averaging over longer periods (i.e and 24 h), hence allows for some meandering of the wind, and this factor may be the reason for the good agreement between the model predictions and field observation However at high Froude numbers the Valley model suggests that the height of the plume and its size over the terrain is the same as if the plume were over level ground Our experiments, along with many others in wind tunnels and full scale, have shown that the height of the plume above the surface of a three-dimensional hill differs considerably from its value over level ground Also the plume depth normal to the surface (a:) increased considerably (Estimates for these changes are given in Section of Part I.) The net effect is that the surface concentration may be increased considerably, in our case by a factor of 12 compared with its value over level ground Acknowledgements-We are grateful to Mr Roger Thompson and Mr Daniel Dolan for help with the photographs, and to Mr Robert Lawson, Mr Lewis Knight, Mr Leonard Marsh, and the Late Mr Kari Kurfis for help with running the experiments Financial support for J.C.R.H was provided through his appointment as a Visiting Associate Professor, Department ofGeosciences, NC State University through EPA Grants 804653 and 805595 REFERENCES Baines P G (1979) Observations of stratified flow past threedimensional barriers J geophys Res 84, 7834-7838 Bass A., Strimaitis D G and Egan B A I19811Potential BOW Turbulent difiusion from a point source in stratified and neutral flows around a three-dimensional hi&-II model for Gaussian plume interaction with simple terrain features Report to EPA under Contract NO 68-02-2759, Research Triangle Park, NC 277 11 Brighton P W M (1978) Strongly stratitied Sow past threed~ensio~ obstacles Q JI R met Sot lt&289-307 Britter R E and Hunt J C R (1984) Diffusion from sources upwind of bluff obstacles in grid turbulence (in nrenaration) Burt E W and Skater H H (1977) Evaluation of the Valley Model AMS-APCA Joint Conf on Appl of Air Poll Meteorol., Salt Lake City, UT CastroI P and Snyder W H (1982) A wind tunnel study of dispersion from sources downwind of thr~d~mensioMl hills Atmospheric Enuironment 16, 18691887 Drazin P G (1961) On the steady flow of a fluid of variable density pasi an obstacle Tellus 13, 239-251 Eaan B A (1975) Turbulent diffusion in complex terrain Workshop on Air Poll Meteorol and Envir ksst%%,Am Met Sot., Boston, MA Egan B A and Bass A (1976) Air quality modeling of effluent plumes in rough terrain Proc 3rd Symp on Atmos Turk Dijiision and Air Quality, Raleigh, NC, Am Met Sot., Boston, MA, pp 484487 Huber A H., Snyder W H., Thompson R S and Lawson R E., Jr (1976) Stack placement in the lee ofa mountain ridge: a wind tunnel study EPA Report No EPA-W/4-76 047, EPA, Research Triangle Park, NC 27711 Hunt J C R., Britter R E and Puttock J S (1979) Mathematical models of dispersion of air pollution around buildings and hills Proc I.M.A Symp on Math Models of Turb Di$I in Enuir., Liverpool, 1978 (edited by Harris C J.), pp 145-200 Academic Press, New York Hunt J C R and Mulheam P J (1973)Turb~ent dispersion from sources near two-dimensional obstacles Fluid Mech 61, 245-274 Hunt J C R., Puttock J S and Snyder W H (1979) Turbulent diffusion from a point source in stratified and neutral flowsaround a three-dimensional hill I Diffusion equation analysis Atmospheric Environment 13, 1227-1239 Hunt J C R., Richards K J and Brighton P W M (1984) Stratified shear flow over low hills II Stratification effects in the outer flow region Q Jf R Met Sot (sub~tt~) Hunt J, C R and Snyder W H (1980) Experiments on stably and neutrally stratified flow over a model threedimensional hill J.Fluid Mech %, 671-704 Hunt J C R Snvder W H and Lawson R E Jr (19781Flow structure and turbulent diffusion around a threedimensional hill: fluid modeling study on effects of stratification I Flow structure Rpt No EPA-m/4-78041, EPA, Research Triangle Park, NC 27711 Khurshudyan L H., Snyder W H and Nekrasov I V (1981) Flow and dispersion of pollutants over two-dimensional hills: Summary report on joint Soviet-American Study EPA Report No EPA-600/4-81-067, EPA, Research Triangle Park, NC 27711 Lavery T F., Bass A., Strimaitis D G., Venkatram A., Greene B R., Drivas P J and Egan A (1982) EPA CompIex Terrain Modeling Program: first milestone report-1981 EPA-600/3-82-036, Research Triangle Park, NC, 27711 Paspuill F (1974) Atmosuheric Diffusion (2nd Ed.) Ellis H&wood, khichester ’ Puttock J S (1979) Turbulent diffusion from sources near obstacles with separated wakes-Pt II Concentration measurements near acircularcylinder in uniform turbulent flow Ar~spberie En~iron~Rt 15, 15-22 Riley J J., Liu, H T and Geller E W (1976) A numerical and experimental study of stably stratified flow around complex terrain, EPA-600/4-76-021, EPA, Research Triangle Park, NC 2771 Rowe R D., Benjamin S F., Chung K P., Havlena J J and Lee C Z (1982) Field studies of stable air flow over and around a ridge Atmospheric Environment 16, 643-653 s I r 2001 Sheppard P A (1956) AirSow over mountains Q JI R met Sot g&528-529 Snyder W H (1979) The EPA Meteorological Wind Tunnel: its design, construction, and operating characteristics EPA~/~79-051, EPA, Research Triangle Park, NC 27711 Snvder W H and Britter R E (1984) Dispersion from &rces upwind of three-dimensional hills (in prep.) Snvder W H Britter R E and Hunt J C R (19801A fluid modeling study of the flow structure and plume impingement on a three-dimensional hill in stably stratified flow Proc Fi$f? Inc Cor$ on It-fad Ewr (edited by Cermak J E.), pp 319-329 Pergamon Press, New York Snyder W H., Thompson R S., Eakridge R E., Lawson R E., Jr., Castro I P., Lee J T., Hunt J C R.and Ogawa Y (1984) The structure of strongly stratified flow over hills: dividing streamline concept J Fluid Me& (sub~tt~~ Turner D B (1970) Workbook of Atmospheric Dispersion Estimates, Office of Air Programs Pub No AP-26 EPA, Research Triangle Park, NC 27711 Weil J C (1983) Application of advances in planetary boundary layer understanding to diffusion modeling Proc Sixth Symp on Turbulence and f@sion, 22-25 Mar Boston, MA, Am Met !Ioc., Boston, MA, pp 42-46 Weil J C., Traugott S C and Wong D K (1981) Stack plume interaction and tlow characteristics for a notched ridge Rpt No PPRP-61, Maryland Power Plant Siting Program, Martin Marietta Corp., Baltimore: MD Wu J (1969) Mixed region collapse with internal wave generation in the density stratified medium J Fluid Mech 35.531-544 APPENDIX: PLUME MEANDER AND THE MEAN CONCENTRATION The following model is constructed to estimate the mean concentration at the stagnation point (N, e HB) when the source is close enough to the centerline that the plume oscillates to and fro across the stagnation point Near the stagnation point ( y B R,, 1x1tg R,), the flow approximates that past an inside comer, so that the streamlines form rectangular hyperbolas and the streamfunction is ti = - 2U,xylR, (Al) Upstream, the str~fu~tion passes through the source at y = ys, so that $ = - U,y, We shall define the laterai displacement of the impingement point y as the y-coordinate of that point on the st~mIine throu~ the source that is nearest to the stagnation point, so that Y: = R,Y,/~ (A2) The fhtctuations in yd shown in Fig 17 clearly have a bimodal form, so that we shag approximate the probability distribution as two delta functions where yb = [(yd -jrd)*]tiz, and the overbar denotes an average From (A2) and (A3) we can estimate the equivalent variations of source offset position corresponding to the fluctuations in yd This simple model, in e&et, assumes that the equivalent source position has two values, g’, Jc2 = 2(yd f YI,)‘/‘R, (A41 In our observations (e.g Fig 17), we found that when y,J& = 0.075, ;id/R, = 0.15, and yi/R, = 0.09, and when y,/R, = 0.038, jr*/& = 0.063 and y$$ = 0.17 WILLIAMH SNYDER and J C R HUNT 2002 Using (A4), this implies for y,/R, = 0.075, y,“/R, for YSI& = 0.038, yf/R, = 0.12 and g*/R, = 0.11 and g2jR, = 0.01; lations for ~(0, y,) from Fig of Part 1, we therefore estimate the mean concentration to be Y(e.y,) = k(6ti9 = 0.02 Thus, we may roughly estimate that for y15 = ‘E jR ye’jR s - +0.06 and g2 = -0.06, which corresponds to yd/Rs = rt 0.17, or about It IO” from the stagnation line Assuming ur/yd is small, and using the theoretical calcu- or $X(without + x(Q~*)]/z meander) = 0.7 Thus, the effect of the plume oscillations is estimated to reduce the mean concentration at the stagnation point by approximately 30 Y$,a value in reasonable agreement with the , observations ... Khurshudyan et al (1981) 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... 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 R, in this case is zero The actual offset... were drawn to fit the data as measured, but a generous allowance was made for scatter in order that they might also satisfy our physical intuition The graph suggests that the largest concentrations