Effect of Vertical Wind Shear on Concentration Fluctuation Statistics in a Point Source Plume

Transcription

1 Boundary-Layer Meteorol (2008) 129:65 97 DOI /s y ORIGINAL PAPER Effect of Vertical Wind Shear on Concentration Fluctuation Statistics in a Point Source Plume Trevor Hilderman David J. Wilson Received: 31 July 2006 / Accepted: 29 July 2008 / Published online: 21 August 2008 Springer Science+Business Media B.V Abstract Measurements of concentration fluctuation intensity, intermittency factor, and integral time scale were made in a water channel for a plume dispersing in a well-developed, rough surface, neutrally stable, boundary layer, and in grid-generated turbulence with no mean velocity shear. The water-channel simulations apply to full-scale atmospheric plumes with very short averaging times, on the order of 1 4 min, because plume meandering was suppressed by the water-channel side walls. High spatial and temporal resolution vertical and crosswind profiles of fluctuations in the plume were obtained using a linescan camera laser-induced dye tracer fluorescence technique. A semi-empirical algebraic mean velocity shear history model was developed to predict these concentration statistics. This shear history concentration fluctuation model requires only a minimal set of parameters to be known: atmospheric stability, surface roughness, vertical velocity profile, and vertical and crosswind plume spreads. The universal shear history parameter used was the mean velocity shear normalized by surface friction velocity, plume travel time, and local mean wind speed. The reference height at which this non-dimensional shear history was calculated was important, because both the source and the receptor positions influence the history of particles passing through the receptor position. Keywords Concentration fluctuations Integral time scale Shear history Wind shear 1 Introduction Hazard assessments from toxic or flammable releases often require a wide variety of source sizes, release rates, atmospheric conditions, and receptor locations to be calculated. At present there is no practical model for concentration fluctuation statistics (intensity, intermittency, time scale) that can be used for the repetitive calculations needed in hazard assessments. T. Hilderman D. J. Wilson Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8 T. Hilderman (B) Coanda Research & Development Corporation, Cariboo Road, Burnaby, BC, Canada V3N 4A3 trevor_hilderman@coanda.ca

2 66 T. Hilderman, D. J. Wilson Whether the concentration exposure is important in terms of toxicity, flammability, or any other adverse effect, most receptors are located less than 2 m above the ground. This region of interest is also the most difficult to predict or measure experimentally because of the close proximity of the fixed surface and the non-homogenous nature of turbulent mixing, even in relatively simple neutrally stable conditions. A secondary motivation for this work was that these concentration fluctuation statistics are necessary as inputs to the stochastic concentration time series model described in Hilderman and Wilson (1999). In this study, we hypothesize that vertical wind shear history is the dominant factor that attenuates the concentration fluctuations near the ground as particles are advected from the source to the receptor. This idea is implicit in all our interpretations of the physics governing concentration fluctuation statistics in a turbulent boundary layer. Essentially, our hypothesis is that all the actual features of near-wall boundary-layer turbulence (mean shear, small turbulence scales, high intensities, no rough-surface transfer boundary condition) modify the concentration fluctuations that would otherwise exist, through the multiplicative effect of a single mean velocity shear correction. The really surprising observation is that the shear effects can be treated by a multiplicative function. The physics behind this deserves attention in future research. The shear history model we propose extends currently available dispersion models, such asthemeanderingplumemodelofgifford (1959) andextendedbysawfordandstapountzis (1986), Sykes (1988)and Wilson (1995), that predict plume statistics in more homogeneous turbulence well above the ground. The algebraic shear history model we present here can be used with any dispersion model that predicts no-shear concentration fluctuation statistics to adjust for the effect of vertical wind shear on intensity i, intermittency factor γ and integral time scale T c of concentration fluctuations in a well-developed rough surface boundary layer. Recent work by Luhar et al. (2000) and Cassiani and Giostra (2002) has been done on modelling higher order concentration statistics in the convective boundary layer with no mean velocity; validated using the laboratory data of Deardorff and Willis (1984). In the present study we exclude the limiting case of a purely convective boundary layer because it has no mean shear history. Many other laboratory and full-scale experiments have collected concentration fluctuation data, but much is of horizontal and vertical profiles measured at some elevated position and typically at the source height. In order to measure the effect of shear history it is important to have good resolution through the vertical extent of the boundary layer. Some past experimental measurements of vertical profiles are the wind-tunnel studies of Fackrell and Robins (1982a), water-channel studies of Bara et al. (1992), and atmospheric studies of Mylne (1993) and Yee et al. (1995). The measurements used in the present study were taken with the linescan laser-induced fluorescence (LIF) technique described in Hilderman and Wilson (2007) and have much better spatial and temporal resolution than any previous experiments. The most challenging aspect of the model development was to find a single physically realistic model that included the effects of receptor position and release height. 2 Information Required for a Concentration Fluctuation Model At a minimum, the following parameters are required to adequately describe concentration fluctuations: Mean concentration C: The mean concentration can be obtained from one of the innumerable dispersion models available. It will be assumed that the mean concentration is known.

3 Effect of Vertical Wind Shear on Concentration Fluctuations 67 Fluctuation intensity i:definedas c 2 i = C 2, (1) where c 2 is the variance of the concentration and c 2 = c rms is the standard deviation or root mean square fluctuation. (The convention used here is c = C + c where c is the instantaneous concentration and c is the fluctuation from the mean C). Conditional (in-plume) fluctuation intensity i p : calculated by excluding the zero concentration intermittent periods, so i p = c 2 p C 2 p, (2) where c 2 p is the conditional concentration variance and C p is the conditional mean concentration. Intermittency factor γ : defined as the probability of the concentration being greater than zero (i.e. the fraction of time during which there is measurable non-zero concentration). The total and conditional fluctuation intensities are related to the intermittency factor by an exact equation, Wilson et al. (1985)(seealsoWilson (1995), Eq. 8), γ = 1 + i 2 p 1 + i 2, (3) Integral time and length scales of concentration fluctuation T c, and L c : the integral time scale T c is the area under the autocorrelation curve for the concentration time series. Following Hinze (1975, pp ), the time scale can be calculated from a time series of data by computing the one-dimensional power spectrum. The zero frequency intercept E c (0) is found by extrapolation and T c = E c(0). (4) 4c 2 This time scale is unaffected by the plume intermittency factor γ, an empirical conclusion, supported by the derivation in the Appendix. Using the frozen turbulence assumption, the integral length scale is just L c = T c U where U is the local mean velocity. There are limits on the applicability of the frozen turbulence assumption, i.e., that u rms /U < 0.2, say. Near the surface, u rms 2.5u, and with U 2.5u ln(z/z 0 ) we expect the frozen turbulence approximation to be strictly applicable only for z 150z 0. Models will be developed here for T c, L c, i, i p and γ in a shear flow boundary layer. 3 Experimental Description All of the measurements used in this study were obtained with linescan laser-induced fluorescence (LIF) optical measurement techniques in the recirculating water channel in the Department of Mechanical Engineering at the University of Alberta. Disodium fluorescein (C 20 H 10 Na 2 O 5 ) dye solutions were injected into either a rough surface boundary layer or a shear-free grid turbulent flow in the 5,240 mm long by 680 mm wide by 470 mm deep test section of the channel. High spatial and temporal resolution concentration measurements

4 68 T. Hilderman, D. J. Wilson grid of 3/4 square stainless steel bars 4 vertical and 4 horizontal unequally spaced fluorescein dye source (elevated jet source shown) linescan camera aligned with laser beam 1024 pixels 12 bits/pixel 500 lines/second water surface channel width 680 mm water depth 400 mm U z h s stainless steel shark s tooth fence expanded metal roughness array x=0 z=0 y axis into page x 150 mw argon-ion laser beam 2750 mm Fig. 1 University of Alberta Mechanical Engineering Department water-channel schematic. The recirculation piping, downstream weir gate, and inlet plenum flow straighteners are not shown. Coordinate system origin is at ground level on the channel centreline at the downstream location of the tracer source. Laser beam diameter is approximately 1 mm and projects into the page were made with a Dalsa model CLC6-2048T 12-bit grey-scale CCD linescan camera along a line illuminated by an argon-ion laser. Each of the 1,024 pixels had a measurement volume of approximately mm 3 and were sampled at 500 samples per second for a total of 500 s per experiment. Figure 1 is a schematic of the water channel that shows a laser line entering the water channel from the side and with the camera on top of the channel to produce a horizontal profile of concentration. The most important shear flow data were produced by reversing the positions of the laser line and the camera and measuring vertical profiles through the dispersing plume. For comparison with full-scale atmospheric measurements, 1 s in the water channel is roughly equivalent to 1 min in the atmosphere. The equivalent full-scale measurement time was approximately 500 min 8 h at a rate of eight samples per second at 1,024 simultaneous points across the plume. Sawford (2004) confirms that atmospheric simulation experiments like this are valid for determining first-order and second-order statistics of scalar dispersion. The corrections required for low Reynolds number (Re) and high Schmidt number experiments with dye in a water channel are small as compared to low Sc gases dispersing in the high Re atmospheric environment and should have little effect on the first-order and second-order statistics of scalar dispersion. 3.1 Flow Fields for Experiments Rough Surface Turbulent Boundary Layer For most experiments, the water channel was configured as in Fig. 1 to produce a welldeveloped rough surface turbulent boundary-layer flow to simulate the atmosphere under neutrally stable conditions. The rough bottom surface was made of nominal 12 mm 18 gauge raised surface stainless steel expanded metal fastened to 6 mm thick acrylic panels. The expanded metal had diamond shaped openings approximately 11 mm wide in the flow direction and 24 mm wide in the cross-stream direction. The raised surface extended 4 mm

5 Effect of Vertical Wind Shear on Concentration Fluctuations 69 above the acrylic panels. Boundary-layer development was accelerated by additional flow conditioning elements placed at the inlet of the channel test section. An array of four horizontal and four vertical 19 mm stainless steel square bars and a 70 mm high trip fence with 40 mm high by 60 mm wide triangular teeth were used to redistribute the flow and generate some mid- to large-scale turbulence (relative to the size of the water channel). A two-component TSI laser Doppler velocimeter (LDV) system was used to make measurements of the velocity profiles in the channel and fine tune the positions of the square bar and trip fence conditioning elements. The cross-stream uniformity of the mean streamwise velocity U was ±5% across the channel. Figure 2a shows a typical vertical profile of the mean streamwise velocity U measured at 3,000 mm downstream of the channel inlet. The source position was 2,750 mm downstream of the channel inlet so these velocity measurements are at a normalized position of x/h = where H is the boundary-layer depth. The logarithmic law fit to the profile is U = u κ ln ( z d z 0 ), (5) (a) (b) H = 400 mm Measured at x/h=0, y=channel centreline x/h=0, y=channel centreline u' rms / U H 0.8 x/h=2.5, y=channel centreline 0.8 v' rms / U H 0.7 log law fit u * = 14 mm/s 0.7 w' rms / U H 0.6 z 0 = 0.52 mm 0.6 d = 1.7 mm 0.5 U H = 232 mm/s U / U H Normalized rms velocity fluctuation (z-d) / H (c) (z-d) / H Measured at x/h=0, y=channel centreline T u U H / H T v U H / H T w U H / H (z-d) / H (d) (z-d) / H Normalized Eulerian Integral Time Scales -uw / U 2 H Normalized Re stress Measurement Position x/h=0, y=channel centreline x/h=2.5, y=channel centreline Fig. 2 Velocity statistics for the rough surface boundary layer shear flow measured relative to the source position, x/h = 0, which was 3,000 mm downstream from the water-channel inlet. (a) Normalized vertical profiles of the mean streamwise velocity U/U H.(b) Normalized vertical profiles of the rms fluctuating velocity components u rms /U H, v rms /U H,andw rms /U H.(c) Vertical profiles of the normalized Eulerian velocity fluctuation time scales, T u U H /H, T v U H /H,andT w U H /H.(d) Vertical profiles of the normalized Reynolds shear stresses uw/u 2 H

6 70 T. Hilderman, D. J. Wilson where u = 14 mm s 1 is the friction velocity, κ = 0.4 is the Von Karman constant, d = 1.7 mm is the zero-plane displacement, and z 0 = 0.52 mm is the roughness length. All z coordinates are measured from the bottom of the roughness (i.e. the top of the expanded metal is at z = 4 mm.) The zero-plane displacement d is a virtual smooth surface position necessary to fit the log law and is a function of the real roughness length and density of the roughness elements. The log law mixing-layer depth H = 400 mm = 769z 0 was the entire depth of the channel. The velocity at H was U H = 232 mm s 1,whichwasusedasa normalizing factor in the plots. The shear is the partial derivative of Eq. 5 with respect to z U z = u κ (z d). (6) In the log law profile, the velocity goes to zero at (z d) = z 0 at which point there is still a finite velocity gradient u /κz 0. The profiles in Eqs. 5 and 6 should only be used down to the zero velocity point z min = z 0 + d 2.2 mm. In the water channel the lowest point at which measurement were taken was z = 6 mm, well above z min. Velocity fluctuations u rms, v rms and w rms are shown in Fig. 2b normalized by U H,and Fig. 2c shows vertical profiles of the Eulerian integral time scale of velocity fluctuations, T u, T v,andt w, all normalized by H/U H = 1.7 s. Figure 2d shows the vertical profile of the uw Reynolds shear stress. This linear profile indicates fully-developed free-surface channel flow with the rough-surface constant stress layer very close to the ground, as expected in a zero-pressure gradient boundary layer in the atmosphere Grid Turbulence as a Zero Shear Reference For comparison purposes, plume dispersion measurements were also made in a shear-free grid-generated turbulent flow. The grid was made of flat stainless steel bars 19.2 mm wide by 5 mm thick with a centre to centre spacing of G = 76.2 mm and a total open area of 56%. The bars were standard stainless steel rolled stock with slightly rounded edges. The grid was positioned 340 mm downstream of the channel inlet and the flow was run 405 mm deep with a U = 200 mm s 1 average speed. The cross-stream variation of the mean streamwise velocity U was at most ±5% if the side-wall boundary layers were neglected. The vertical fluctuations w rms were approximately 95% of the streamwise fluctuations u rms indicating slight anisotropy in the flow. As expected for grid turbulence, the turbulence intensity decays with downstream distance as shown in Fig. 3a. The power-law curve plotted on the figure is the best fit to the power-law decay of grid turbulence using Saffman s invariant, see Hinze (1975, pp. 217, ). As expected the decay of grid-turbulence intensity is a power law. The relationships given in Hinze (1975, Eqs and 3-186) were reformulated in terms of normalized downstream distance from the source x/g = Ut t /G, ( ) u 0.6 rms x U = 0.3 G + 17, (7) with the constants 0.3 and 17 fitted to the present data. To simulate dispersion in homogeneous turbulence, the dye source at x/g = 0 was placed 19.1G downstream from the grid where the turbulence intensity was about 5%. At the farthest downstream measurement position, x/g = 19.7, the turbulence intensity decayed to about 3%.

7 Effect of Vertical Wind Shear on Concentration Fluctuations 71 (a) Turbulence Intensity (%) grid position x/g = source position x/g = 0 u'/u on channel vertical and cross-stream centreline z/g = 2.6, y/g = 4.7 w'/u on channel vertical and cross-stream centreline z/g = 2.6, y/g = 4.7 u' rms /U = 0.3(x/G + 17) -0.6 Best Fit of Power Law Decay of Grid Turbulence using Saffman's Invariant (b) Normalized Eulerian Integral Time Scales x/g Normalized Downstream Distance from Source grid position x/g = source position x/g = 0 T u U / G on channel vertical and cross-stream centreline z/g = 2.6, y/g = 4.7 T w U / G on channel vertical and cross-stream centreline z/g = 2.6, y/g = 4.7 theoretical integral scale T u from best fit power law decay T u U/ G = 0.1(x/G + 17) 0.4 theoretical integral scale T w from best fit power law decay T w U/ G = 5(x/G + 17) x/g Normalized Downstream Distance from Source Fig. 3 Grid turbulence velocity statistics. (a) Turbulence intensity decay with normalized downstream distance x/g along the centreline of the channel. Best fit power-law decay of turbulence intensity based on Saffman s invariant. (b) Normalized Eulerian integral timescales of velocity fluctuation in the streamwise T u U/G and vertical T w U/G directions along the centreline of the channel compared to T u U/G calculated using the best fit power-law decay of grid turbulence based on Saffman s invariant The normalized Eulerian time scale of velocity fluctuation for the streamwise component was approximately T u U/G = 0.4 and for the vertical component T w U/G = 0.2 asshown in Fig. 3b. The two curves on this plot are the theoretical streamwise time scale calculated using the grid power-law decay as in Eq. 7, ( ) 0.4 T u U x G = 0.1 G (8) The curve for the vertical time scale T w is one half of that given in Eq. 8.

8 72 T. Hilderman, D. J. Wilson The fit to theory with the exponent of 0.4 fixed by Saffman s invariant was not as good for time scales as for the turbulence intensity decay, but the general shape is correct and the ratio between the measured streamwise and vertical scales is almost exactly the expected factor of 2.0 that would occur in homogeneous isotropic turbulence. For this study the velocity fluctuation integral time scales were taken as the average values, in normalized terms T u U/G = 0.39, T v U/G = T w U/G = Tracer Sources Three different dye sources were used as shown in Fig. 4: 1. Horizontal jet: 4.3 mm outside diameter and 3.25 mm inside diameter stainless steel tube, 38 mm long was suspended from above the channel by a streamlined support. In normalized units the source diameter d s 6z 0 08H in the shear flow and d s 4G in grid turbulence. In grid turbulence, the source was placed at the centre of the channel at z = 200 mm = 2.6G above the channel bottom, and in the shear flow the source was placed at height h between 7 and 50 mm ((h d)/h = 13 to 0.12 or (h d)/z 0 = 10 93) above the surface depending on the experiment. With the small diameter and low flow rates the jets from the source were laminar, Re = U source d s /ν 600, U source was equal to the cross-flow U at h = 50 mm. 2. Vertical jet at ground level: 3.25 mm inside diameter stainless steel tube was set flush with the lower boundary (d s 6z 0 08H). To prevent dye from becoming trapped in the roughness elements the expanded metal was removed from an area 25 mm on all sides and 100 mm downstream of the source. The source flow rate was the same as for the horizontal jets and produced a laminar jet with a mean speed equal to the cross-flow speed at (z d)/h = 0.12, Re 600 based on source diameter, U source was equal to the cross-flow U at h = 50 mm. 3. Large ground level source: 11 mm inside diameter tube was set flush with the lower surface (d s 21z 0 28H). As with the vertical ground level jet the expanded metal was trimmed away 25 mm on all sides and 100 mm downstream of the source. The source flow rate was the same as the other two sources, Re 175 based on source diameter and U source for this case was 8.7% of the flow velocity for the vertical and horizontal jet sources. The sources were placed 2,750 mm downstream of the channel inlet in the shear flow and 1,800 mm downstream of the grid. The source position is used as the zero origin for the x-coordinates. The vertical ground level sources had very low momentum with insignificant plume rise. If modelled at 1:1,000 scale the full-scale equivalent source sizes were 3 11 m at the source and effectively 2 3 times larger than this after entraining sufficient fluid to take on the turbulent structure of the flow field. Measurements were taken at x/d s > 150 for the jet sources and x/d s > 50 for the large ground level source. At this downwind point the dilution was at least 100:1, which allowed the tracer-marked fluid to take on the turbulent structure of the cross-flow. There was little measurable effect of release rate in our experiments. Approximately 150 linescan datasets were collected with 84 of the datasets selected for more in-depth analysis. Only a few samples of these data will be shown here, but the trends and observations discussed apply to all of the data collected. Accurate measurement of concentration fluctuation statistics in the outer edges of the plume was limited by the total measurement time see Pasquill and Smith (1983, pp and Fig. 2.2), for discussion of sampling time effects on variance. In the laboratory, flow fields

9 Effect of Vertical Wind Shear on Concentration Fluctuations 73 (a) dye supply tube streamlined fairing flow direction 38 mm long 4.3 mm OD 3.25 mm ID stainless steel tube fluorescein dye jet Horizontal Jet Source - Side View (b) expanded metal roughness flow direction 3.25 mm ID or 11 mm ID source flush with ground roughness removed 25 mm from source on sides, upstream and 100 mm downstream of source. Ground Level Sources - Top View Fig. 4 Fluorescein dye sources. (a) Side view of elevated horizontal jet sources. Source was suspended from above the channel. (b) Top view of ground level sources. Expanded metal roughness was removed from the immediate area of the source and dye supply lines were underneath the acrylic panel below the roughness. The large (11 mm internal diameter) ground level source was changed to the small ground-level source by inserting a plug with a 3.25 mm internal diameter hole for the small source and dispersing plumes can be made statistically stationary and sampled for relatively long periods of time, but there are still practical limitations. In the present experiments, samples were collected for 500 s at each measurement position at 500 samples per second for a total of 250,000 samples per pixel. This was sufficient to resolve profiles of mean concentration, variance and other statistics out to approximately 2σ or 3σ from the source centreline in the y or z directions. As γ decreases near the edges of the plume, the effective non-zero

10 74 T. Hilderman, D. J. Wilson sample size decreases and the measured variance, especially the in-plume conditional variance, decreases. In all cases discussed here, the data used were limited to cases where γ 0.1 to ensure sufficiently long non-zero time series. At γ = 0.1, there are only 50 s out of 500 s that have useful data, which is about 200 non-zero integral time scales out of a total of 2000 concentration integral time scales of data. 4 Predicting Concentration Fluctuations In a boundary layer, the vertical mean concentration profile is usually modelled without reference to shear by assuming a zero-mass flux at the ground. On rough surfaces such as ours, with densely spaced low roughness elements so that d z 0, experiments show that the concentration variance and fluctuation time scale do not go to zero at the top of the roughness elements. Our central hypothesis is that the most important parameter for the vertical variation of concentration fluctuation intensity, intermittency, and time scale in a boundary layer is the non-dimensional velocity shear history of the plume from source to receptor. The objective is to develop a practical operational fluctuation model that can be used for rapid repetitive concentration calculations in existing regulatory dispersion models. An operational model is distinctly different from a research model, which often requires input parameters not readily available outside the laboratory. The operational concentration fluctuation model in Wilson (1995, chap.10) was used as a starting point; this model accounts for some of the effects of shear flow in the equation used to predict the centreline source-height fluctuation reference level i h, but it does not include shear effects off the plume axis, e.g. at ground level. Figure 5 graphically shows the effect of shear near ground level using images constructed from the linescan data. For each of two downstream positions (x/h = 1.25 and x/h = 3.75) six vertical profiles were measured. Three source types, the small vertical ground level jet, the horizontal jet at a high elevated position (z d)/h = 0.12 and the horizontal jet near the ground at (z d)/h = 13 were used with tracer flow rates of 0.7 and 1.5 ml s 1. All 12 profiles were used to fit the model, but only four cases are shown in the figures. There was no significant difference between the two ground level sources or the two source flow rates. As Fig. 6 shows, the release height had little effect on plume spread in the boundary layer. The spread in grid turbulence is much smaller because the velocity variances were much smaller in grid turbulence. This is useful for testing the ability of the shear history model to predict statistics in two flows that have very different scales and turbulence levels. 4.1 Non-dimensional Shear History S The mean velocity strain rate, vertical shear U/ z, has the units of s 1 so an appropriate time scale, T S, is required to non-dimensionalize. Using the frozen turbulence assumption, the time scale T S can be reformulated as a length scale L S divided by a velocity U. The non-dimensional shear history S at height z is then ( ) U S = T S = L S z U ( ) U. (9) z Assuming that shear history is the same across the plume in the y direction, L S = σ z is the logical choice for the length scale in Eq. 9. Near the ground in neutrally stable conditions w rms 1.3u (see Kerschgens et al. 2000), so that σ z w rms t t 1.3u t t where w rms is the

11 Effect of Vertical Wind Shear on Concentration Fluctuations 75 Fig. 5 Linescan data visualization at x/h = 1.25 downstream from (a) ground level small vertical jet source, with concentration integral time scale T c = 0.16 s. (b) elevated (h = 50 mm) horizontal jet source shown as solid line, with T c = 0.10 s. Image was constructed by converting linescan data to a grey scale bitmap and placing consecutive linescans side by side. The horizontal axis in the image is time and the vertical axis is vertical position in the water channel. Total elapsed time is 2.0 s of data at 500 samples per second. Source heights h and vertical plume spreads σ z are shown to scale on the left-hand side of the images root-mean-square vertical velocity fluctuation and travel time t t is defined as t t = x/u, with U a function of z used to account for source-to-receptor shear history effects. Making these substitutions reduces Eq. 9 to or in terms of downstream distance S = 1.3u t t U S = 1.3u x U 2 ( ) U, (10) z ( ) U. (11) z The factor of 1.3 in Eqs. 10 and 11 indicates that S is defined in terms of the vertical velocity fluctuation component w rms, but expressed in terms of u for convenient calculations in shear flow. The shear history factor S can be expressed in terms of the along-wind shear-induced plume spread σ x,shear. Smith s (1965) exact solution for a linear mean velocity profile with constant shear is, σ x,shear = σ zt t 12 U z, (12)

12 76 T. Hilderman, D. J. Wilson Normalized Horizontal Plume Spread σ y /H or σ y /5G (a) Grid Turbulence Large Ground Level Source Small Ground Level Vertical Jet Horizontal Jet at Ground Level Elevated Horizontal Jet h=25mm Elevated Horizontal Jet h=50mm Normalized Downstream Distance x/h or x/5g Normalized Vertical Plume Spread σ z /H or σ z /5G (b) Grid Turbulence Small Ground Level Vertical Jet Horizontal Jet at Ground Level Elevated Horizontal Jet h=50mm Normalized Downstream Distance x/h or x/5g Fig. 6 Plume spreads measured in the water channel as a function of downstream distance x/h for shear layer thickness H = 400 mm and x/5g for zero-shear grid spacing G = 76.2 mm. (a) Normalized horizontal plume spread, σ y /H and σ y /5G (b) normalized vertical plume spread, σ z /H and σ z /5G (see Wilson 1981), and substituting σ z w rms t t and t t = x/u in Eq. 12, weobtain 12σx,shear S =, (13) x and our shear-history term becomes simply the normalized along-wind plume spread at each height z. This is a significant result, because it illustrates that S, like σ x,shear,isanintegrated history over the plume travel time. Equation 11 can also be rewritten in terms of the roughness length z 0 and displacement height d of the log-law velocity profile instead of the friction velocity u. By substituting the log-law velocity profile Eq. 5 for U and the shear profile Eq. 6 for U/ z, then Eq. 11 becomes

13 Effect of Vertical Wind Shear on Concentration Fluctuations κ ( ) x z 0 S = ( ) ( ). (14) z d z d ln 2 z 0 z 0 Note that Eq. 14 shows that shear history S is independent of friction velocity u. Equation 10 was used as the working equation for all calculations and figures that follow. 4.2 Algebraic Model for Shear History Effects Using the pseudo-meandering plume model, profiles of mean concentration and total second-order moments (mean squared plus variance) are both approximately Gaussian in shape with different spreads for the mean and total second moments. These Gaussian profiles also agree well with the data. Figures 7 and 8 show how well a simple meandering plume model predicts intensity on and off the plume axis for both a boundary layer in Fig. 7 and no-shear grid turbulence in Fig. 8. The lines are from Wilson (1995, Eq. 6.8) for dispersion in homogeneous turbulence with a uniform wind U, the no-shear case, ( )] ino-shear = (i h,no-shear [exp 2M intensity (z h) ) 2σz 2 + y2 1+2M intensity 2σy 2, (15) where ih,no-shear 2 is the fluctuation intensity squared at the source height h on the cross-stream centreline of the plume (i.e. at y = 0) and M intensity is the two-dimensional pseudo-meander parameter defined in Wilson (1995, Eq. 6.10). Pseudo-meandering refers to a profile where the meander ratio M is calculated from the fluctuation intensity i and not evaluated directly from estimates of plume meander spread σ m. This equation was derived from the higher moment concentration relations developed by Sawford and Stapountzis (1986), M intensity = i 2 h,no-shear + (i 4 h,no-shear + i 2 h,no-shear )1/2. (16) In a no-shear flow, this simple pseudo-meandering plume model works exceptionally well. As expected, mean wind shear U/ z attenuates concentration fluctuations near the ground and significant modifications to the pseudo-meandering plume model are needed. Strictly speaking, Gaussian profiles of mean concentration and meandering plume intensity in (15) imply homogeneous turbulence with a constant crosswind and vertical diffusivity. Yet Gaussian profiles are very widely used in regulatory models to predict mean concentrations in the atmospheric boundary layer where the turbulence near the surface is certainly not homogeneous in variance or scale. Meandering plume models give good estimates of crosswind concentration fluctuation intensity profiles in non-homogeneous turbulence of full-scale and model atmospheric boundary layers. In both mean concentration and intensity profiles the inhomogeneous turbulence field is captured within a single parameter, the plume spread σ z or σ y. Once the plume spreads are known (a non-trivial task in the real atmosphere, which is non-stationary as well as inhomogeneous) all that is needed is to specify how far off the plume axis the receptor is, and how the effect of the ground surface is taken into account. Our shear history model makes the further assumption that the solution for dispersion in shear flow can be represented in an algebraic variables-separable format, with the noshear solution adjusted by a single multiplicative term that accounts for the shear history from source to receptor. In effect this is an implicit integral theory for plume behaviour,

14 78 T. Hilderman, D. J. Wilson (a) i Concentration Fluctuation Intensity Shear Flow Boundary Layer Horizontal Jet (h-d)/h=0.12 x/h = 2.5 (z-d)/h=11 (z-d)/h=58 (z-d)/h=0.12 Meandering Plume No Shear Model (z-d)/h=11, σ y /H=0.19 i y=0 =0.48 (z-d)/h=58, σ y /H=0.20 i y=0 =0.97 (z-d)/h=0.12, σ y /H=0.21 i y=0 = y / σ y Cross-stream Position (b) i Concentration Fluctuation Intensity Shear Flow Boundary Layer Horizontal Jet (h-d)/h=13 x/h = 2.5 (z-d)/h=11 (z-d)/h=58 (z-d)/h=0.12 Meandering Plume No Shear Model (z-d)/h=11, σ y /H=0.20 i y=0 =0.46 (z-d)/h=58, σ y /H=0.21 i y=0 =0.72 (z-d)/h=0.12, σ y /H=0.22 i y=0 = y / σ y Cross-stream Position Fig. 7 Samples of typical cross-stream profiles of concentration fluctuation intensity i compared to Eq. 15 at x/h = 2.5 downstream of the source (a) horizontal jet above the ground, (h d)/h = 0.12, in shear flow (b) horizontal jet at ground level, (h d)/h = 13, in shear flow in contrast to describing the plume by a hierarchy of partial differential conservation equations that are then solved numerically for mean, variance, and scale of concentration. Our algebraic approach has the important advantage of allowing shear effects to be incorporated into the in-plume conditional fluctuation intensity i p (excluding periods of zero concentration) for which there is no differential conservation equation; and to find the fraction of time γ that the concentration is non-zero. Both i p and γ are essential to estimating probabilities of exceedance of a fixed concentration threshold. An additional restriction on our shear history model is that S goes to zero in grid turbulence, strongly convective atmospheric turbulence, or at a vertical position well away from the influence of the wall where z z 0. Any relationship derived must reduce smoothly to

15 Effect of Vertical Wind Shear on Concentration Fluctuations 79 (a) 8 i Concentration Fluctuation Intensity Grid Turbulence Horizontal Jet Meandering Plume No Shear Model x/g = 13.1 z/g=0=channel centreline z/g=0=channel centreline σ y /G=0.36 i y=0 =3.17 z/g=0.3 z/g=0.3, σ y /G=0.36 i y=0 =3.64 z/g=0.5 z/g=0.6, σ y /G=0.36 i y=0 = y / σ y Cross-stream Position (b) 8 i Concentration Fluctuation Intensity Grid Turbulence Horizontal Jet z/g = 0 = source centreline x/g=6.6 x/g=13.1 x/g=19.7 Meandering Plume No Shear Model x/g=6.6 σ y /G=0.20 i y=0 =2.55 x/g=13.1, σ y G=0.36 i y=0 =3.17 x/g=19.7, σ y /G=0.50 i y=0 = y / σ y Cross-stream Position Fig. 8 Grid turbulence concentration fluctuation intensity i profiles for a horizontal jet source on the waterchannel centreline compared to the no-shear model, Eq. 15. (a) Measured at x/g = 13.1 downstream at the vertical source centreline on and off the source centreline. (b) Measured at various downstream distances on the source centreline the no-shear case. With this constraint in mind, the proposed universal shear history function is shear statistic no-shear statistic = (1 + B 1S avg ) B 2, (17) where it is possible that the constants B 1 and B 2 in Eq. 17 are functions of other variables such as source size d s, surface roughness z 0, and so on. For the range of variables in our study, we found that only the receptor position and source position shear history S values were needed to correlate the results, and no obvious dependence on source size or jet momentum

16 80 T. Hilderman, D. J. Wilson was evident. All the statistics were found to be well-represented by a single universal value of B 1 = 5, and this value will be used whenever we write the shear history function in Eq. 17. The constant B 2 was allowed to change for each type of fluctuation statistic (i.e. i, T c,ori p ). S avg is the average of the non-dimensional shear history at the receptor height z ref and at the source height h ref, S avg = S z ref + S href, (18) 2 and as illustrated in Fig. 9, both the source and the receptor positions are important. There are several other possible weightings of S zref and S href, for example Gaussian weighting or Lagrangian particle tracking weighting, but the simple arithmetic averaging in Eq. 18 works well. The source supplies the tracer material and projects a downstream zone of influence. This is the effect captured by using a source reference height h ref. Similarly, the receptor position z ref projects an upstream zone of sensitivity. A particle of fluid directly above the receptor has a near-zero probability of passing through the receptor position, but as we examine positions farther upstream it becomes more likely that material from anywhere in the plume could eventually pass through the receptor position. The overlap between these two regions is the relevant value for any receptor position of interest. In Wilson (1981), the approach for computing advection velocities and along-wind dispersion in a shear flow was to calculate an effective height at which to evaluate these parameters. For a ground-level source with h = 0, the effective advection velocity needed to obtain the true travel time is the local velocity at the receptor x location measured at z = 0.17σ z. The effective height was found to be the source height plus some fraction of the vertical plume spread σ z, receptor zone of sensitivity mean velocity and turbulence in this overlap region can affect concentration statistics at the receptor source zone of influence wind speed U source du dz du dz receptor source range of shear affecting sourcereceptor pair 0 X receptor Fig. 9 Physical example of the effects of shear history from the source and the receptor position. The source projects a downstream zone of influence as all material measured at any receptor must come from the source. The receptor projects an upstream zone of sensitivity because the probability of a particle of the source material getting to the receptor depends on the path it follows. The overlap region is the location of the source material most likely to travel to the receptor and the shear in this region is the cause of the shear attenuation at the receptor

17 Effect of Vertical Wind Shear on Concentration Fluctuations 81 z ref = z + B 3 σ z. (19) An offset of some small fraction of σ z also prevents S as U 0atz d = z 0 for a log-law profile (see Eq. 5 and Fig. 2a). Variations in source size are automatically included in the shear history S through the variable σ z. In Eq. 19 we found that setting B 3 = 0.1 gave good agreement with experiment, z ref = z + 0.1σ z, (20) h ref = h + 0.1σ z. (21) With Eqs the modelled values for i, i p, γ and T c typically were within ±20% of the experimental values. It is important to note that our wind shear history model for the reduction of concentration fluctuation intensity (see Sect. 5.1 for more detail) is quite different from models that use differential equations of mass conservation and differential equations for the kinetic energy (velocity variance) budget and scalar fluctuation (concentration variance) budget to estimate the downwind evolution of concentration fluctuation variance and mean concentration separately. Relatively small absolute percentage errors in predicting the separate concentration mean and variance can produce large errors in the intensity i. With this in mind, we chose to predict directly the effect of shear history on intensity. The purpose of predicting intensity i rather than predicting the variance c is that small absolute errors in mean and fluctuating components in the tails of the profile are often not easily visible in curves fitted to graphs, but lead to large percentage errors in i, which is the relevant variable for estimating peak concentration probability from the pdf. 5 Operational Model With Effects of Wind Shear History 5.1 Predicting Vertical Profiles of Fluctuation Intensity i The fluctuation intensity for all positions across the plume is a function of the plume source height and cross-stream centreline fluctuation intensity i h. Under no-shear conditions, the pseudo-meandering plume model of Wilson (1995) in Eq.15 accurately predicts off-axis profiles of intensity i no-shear through the entire plume in both the cross-stream (y) andvertical (z) directions as demonstrated by tests in grid turbulence shown in Fig. 8. In Fig. 8 the no-shear theory uses the measured value of i h,no-shear from the grid turbulence data to normalize the model Eq. 15. This forces agreement between experiment and theory on the plume axis y = 0. The measured intensities were all well-fit with a constant value of B 2 = 1/3 forthe exponent in Eq. 17, and with the universal constant B 1 = 5, we have i shear = i no-shear ( 1 + 5Savg ) 1/3. (22) Comparison of the shear-effect model with experiment requires that the hypothetical no-shear fluctuation intensity i h,no-shear be known so that Eq. 15 can generate a no-shear profile that is then adjusted for shear history using Eq. 22. Figure 10 shows the measurements from the water channel versus the shear-history correction of Eq. 22 on the no-shear model of Eq. 15. The no-shear source-height fluctuation intensity needed in Eq. 15 for comparison with the water-channel experimental data was back-calculated at source height from the shear effects model in Eq. 22, i h,no-shear =

18 82 T. Hilderman, D. J. Wilson i h,shear,measured (1 + 5S avg ) 1/3. This normalization forces the shear-corrected curve through the experimental data at source height. Even considering this forced fit at source height, the agreement with measurements is remarkably good. Clearly, wind shear has a strong influence on concentration fluctuation intensity and must be taken into account. The agreement between the model predictions and experiment shown in Fig. 10 is typical of the results for all 12 profiles taken for the five source configurations (the horizontal jet source was placed at two heights). In Fig. 11, which shows cross-stream profiles, the poorest agreement between theory and experiment for all 12 measured profiles occurred for the highest elevated source. In this worst case, the measurements of total concentration fluctuation intensity i were within 20%, and the conditional in-plume intensity i p within 50%, of the model predictions out to distances of y/σ y 2. Where there is error, the shear-effect attenuation (a) (z-d)/h Normalized Vertical Position Ground Level Source Data Shear Model No Shear shear no shear shear no shear i Concentration Fluctuation Intensity (b) (z-d)/h Normalized Vertical Position Elevated Source (h-d)/h=0.12 Data Shear Model No Shear shear no shear shear no shear source height i Concentration Fluctuation Intensity Fig. 10 Vertical profiles of fluctuation intensity i compared to the shear model Eq. 22 and the no-shear model (15). (a) Ground level source (b) Elevated horizontal jet source (h d)/h = 0.12

19 Effect of Vertical Wind Shear on Concentration Fluctuations 83 Fig. 11 Samples of the best, worst and typical agreement of measured cross-stream profiles with pseudo-meandering plume theory of total concentration fluctuation intensity i, conditional concentration fluctuation intensity i p and intermittency factor γ. The shear model curves are calculated from Eqs. 22, 27 and 3 (a) Best agreement low elevated horizontal jet source (h d)/h = 58 at x/h = 2.5 measured near the ground at (z d)/h = 11. (b) Worst agreement high elevated horizontal jet source (h d)/h = 0.12 at x/h = 1.25 measured near the ground at (z d)/h = 11. (c) Typical agreement large ground level source at x/h = 2.50 measured near the ground at (z d)/h = 11 (a) i, Concentration Fluctuation Intensity i p Conditional Concentration Fluctuation Intensity (b) i, Concentration Fluctuation Intensity i p Conditional Concentration Fluctuation Intensity Shear Flow Boundary Layer Elevated Horizontal Jet (h-d)/h = 58 x/h = 2.5 (z-d)/h = 11 i, data i, shear model i p, data i p, shear model γ, data γ, shear model y / σ y Normalized Cross Stream Position γ, data γ, shear model i, data i, shear model i p, data i p, shear model Shear Flow Boundary Layer Elevated Horizontal Jet (h-d)/h = x/h = 1.25 (z-d)/h = y / σ y Normalized Cross Stream Position γ Intermittency Factor γ Intermittency Factor (c) i, Concentration Fluctuation Intensity i p Conditional Concentration Fluctuation Intensity Shear Flow Boundary Layer Large Ground Level Source x/h = 2.5 (z-d)/h = 11 i, data i, shear model i p, data i p, shear model γ, data γ, shear model y / σ y Normalized Cross Stream Position γ Intermittency Factor model tends to overestimate the fluctuation intensity near the ground, which will produce larger peak concentrations and hence conservative errors in estimating adverse toxic effects of an exposure.

20 84 T. Hilderman, D. J. Wilson 5.2 Predicting Along-Wind Development of Source Height Intensity i h For no-shear homogeneous turbulence far from surface interactions, explicit algebraic equations for i h,no-shear have been developed from meandering plume models, and from the concentration variance c 2 budget equations integrated across and then along the plume. The eddies containing non-zero concentration are assumed to lie within the inertial subrange of the velocity fluctuation spectrum, E v ( f ) v 2 f 5/3 and E w ( f ) w 2 f 5/3,where f is the fluctuation frequency. Fackrell and Robins (1982b) used a meandering plume model to predict the effect of source size on their measured i h wind-tunnel data, while Sykes (1988) obtained an explicit solution for i h,no-shear using a meandering plume model. Fackrell and Robins (1982a) measured most of the terms in the variance budget equation and found that the advection of concentration variance is essentially balanced by the dissipation of the variance beyond a fairly short distance downstream of the source. This advection dissipation balance occurs for most of the travel distance of the plume for both a ground level and elevated source. Assuming Gaussian profiles for the mean C and variance c 2 across a plume in a uniform flow with mean velocity U, and dispersion with plume scales that lie within the inertial subrange of velocity fluctuations, Wilson (1995) usedaninte- grated advection dissipation balance in uniform mean flow (no-shear) with homogeneous turbulence, to obtain, ( ) σe B 4 i h = ( σe L w,ref L w,ref ) ( ) 2/3 3/2, (23) ds + B 5 L w,ref where d s is the source size, and the effective no-shear plume size was defined as σ e,no-shear = (σ y σ z ) 1/2 ; with which Eq. 23 yields i h,no-shear. The inertial subrange is not observed at positions close to the surface in a laboratory scale boundary layer, but as we shall see, the equation fits the data quite well here too. This is exactly the same result obtained by Sykes (1988) from his meandering plume model, with the exception that he used σ e = σ y and L v,ref instead of L w,ref because his plume meandered in the crosswind y direction. Wilson (1995, Eq. 6.4) used Eq. 23 for dispersion in shear flow by applying an empirical shear-history adjustment to the effective plume size, σ e, by adding a multiplicative factor proportional to (σ x,shear /σ z ) 1/2. We propose a similar type of adjustment to σ e using the shear history measured at source height where Eq. 23 is valid, ( ) 1/2 σy σ z σ e = (1 + B 6 S h ). (24) L w,ref Equation 23 with 24 reduces smoothly to the no-shear case for S h = 0. The value of the empirical constant B 6 sets the magnitude of the shear-history correction, much like B 1, while the constant B 5 determines the flow direction position (in terms of σ e ) of the maximum i h. Wilson (1995) used the wind-tunnel data of Fackrell and Robins (1982a) to set the values of B 5 = 3, and we will use this same value in our new model. The constant B 4 = 9 was set to fit the magnitude of the intensity maximum measured for our no-shear grid turbulence data, while the shear-history constant B 6 = 20 was chosen to obtain a best fit to the ground-level source data. Figure 12a shows the source centreline fluctuation intensity