A Laboratory Study of the Urban Heat Island in a Calm and Stably Stratified Environment. Part II: Velocity Field

Transcription

1 1392 JOURNAL OF APPLIED METEOROLOGY A Laboratory Study of the Urban Heat Island in a Calm and Stably Stratified Environment. Part II: Velocity Field JIE LU ANDS. PAL ARYA Department of Marine, Earth and Atmospheric Sciences, North Carolina State University, Raleigh, North Carolina WILLIAM H. SNYDER* AND ROBERT E. LAWSON JR.* Atmospheric Sciences Modeling Division, Air Resources Laboratory, National Oceanic and Atmospheric Administration, Research Triangle Park, North Carolina (Manuscript received 3 June 1996, in final form 8 January 1997) ABSTRACT A fully turbulent, low-aspect-ratio buoyant plume with no initial momentum under calm and stably stratified conditions is produced in a convection tank. The plume is generated by a circular heat island at the bottom of the tank. Two analytical models, a bulk convection model and a hydrostatic model, are developed to formulate similarity relations for the low-aspect-ratio plume. The convective velocity scale w D, suggested by the analytical models, is used as the similarity parameter for both the mean velocity and standard deviations of velocity fluctuations. The normalized standard deviations of horizontal and vertical velocities agree with each other for two heating rates, as well as with field observations in the center of Sapporo, Japan. The suggested scaling and empirical relations based on our experimental results may be applied to the velocity fields of other low-aspectratio plumes in calm and stably stratified environments. Further investigations are recommended to confirm the results of the current study. 1. Introduction A thermal plume is generated by an underlying heat island in the form of an area source. If the heating is confined to a finite area, a vertical thermal plume and associated circulation will develop due to the temperature (density) difference between the heat source and its environs. The plume stops rising as the temperature difference between the plume and its ambient vanishes due to the entrainment or mixing of fluid from a stable environment. Therefore, the height of a plume z i is determined by the strength and size of the heat source, as well as the ambient stratification. As discussed in Part I, the strength of an area heat source may be characterized by the average surface heat flux, and the size of a circular heat source is represented by its diameter D. The aspect ratio of a plume is defined as the plume height-to-diameter ratio z i /D. Thermal plumes can be classified as having high aspect ratios when z i /D k 1, * On assignment to the National Exposure Research Laboratory, U.S. Environmental Protection Agency. Corresponding author address: Dr. William H. Snyder, Mechanical Engineering Department, University of Surrey, Guildford, Surrey GU2 5XH, United Kingdom. w.snyder@surrey.ac.uk intermediate aspect ratios when z i /D 1, and low aspect ratios when z i /D K 1. A good example of a high-aspectratio plume is a point-source plume, and an example of a low-aspect-ratio plume is one created by an urban heat island at night. The temperatures in cities almost always exceed those of the rural surroundings because concrete buildings and paved streets in cities absorb more heat from solar radiation in the daytime, the building s heating and cooling systems, factories, industrial waste gases, and traffic emissions than do the surrounding vegetative surfaces. Thus, at night, because of the larger heat capacity of the city structures, and because of the energy released from ongoing human activities, the city temperatures decrease more slowly than do those of the vegetated rural areas. For this reason, a temperature difference between the urban and rural environments occurs at night; it is referred to as an urban heat island. This phenomenon and its associated circulation are found to be most intense at nighttime under clear skies and weak ambient winds. Previous field experiments and observations (see Part I) indicate that the urban heatisland-induced circulation comprises a buoyant thermal plume over the urban area, a convergent horizontal flow near the surface, a divergent flow aloft, and subsiding motion over the rural environs. With a typical plume height of 200 m at night over the center of a midsize city (10-km diameter), the aspect ratio is about 0.02, which falls in the low-aspect-ratio category.

2 OCTOBER 1997 LU ET AL The high-aspect-ratio plume theory has been well developed in the past. Morton et al. (1956) proposed a model for a thermal plume rising in a stratified environment (hereafter referred to as the MTT model) for such plumes, assuming similarity of velocity profiles in the far-field region. Using the hypothesis that the rate of entrainment at the edge of the plume is proportional to the centerline vertical velocity at the same height, the bulk conservation equations for mass, momentum, and buoyancy were solved to yield the vertical velocity and temperature fields in the far-field similarity region, as well as the plume height. Briggs (1969, 1975) formulated a similar model for a buoyant plume in a stably stratified environment. The MTT model and the Briggs model, referred to as classical high-aspect-ratio plume models, have been verified by extensive experimental data. The flow dynamics change dramatically as the geometry changes from a high-aspect-ratio plume to a lowaspect-ratio plume. For a low-aspect-ratio plume, there is little entrainment or mixing of environmental fluid with the plume, and the similarity region is absent. The classical thermal plume model is not applicable due to the invalidity of its basic hypothesis. Numerical simulations of fire plumes by Heikes et al. (1990) have indicated that plumes from large (radius greater than 5 km) and small fires were fundamentally different because there was significantly less entrainment of ambient air into large fire plumes than into small ones. They found that the plume rise from a small area fire was governed by the total heat release and entrainment, whereas that from a large area fire was determined primarily by the heating rate per unit area and the atmospheric stability a large fire plume has a diameter comparable to the inversion height. A fully turbulent (Reynolds number independent), low-aspect-ratio buoyant plume with no initial momentum is produced under calm and stably stratified conditions in our experiments. The physical model was established in the convection tank of the Fluid Modeling Facility of the U.S. Environmental Protection Agency. The plume was generated by a circular heat island at the bottom of the tank. Because of fundamental differences between high- and low-aspect-ratio plumes and the lack of a completely satisfactory theory or model for low-aspect-ratio plumes, particularly for the velocity fields, the current study is focused on expressing the velocity fields of low-aspect-ratio plumes above heat islands using two simple theoretical models and verifying the predictions with data from laboratory experiments. The empirical relationships obtained in Part I are derived in this part through the two theoretical models. The experimental results of temperature fields, mixing heights, and heat-island intensities for the same realizations of the velocity fields have been reported in Part I. Therefore, only the velocity fields associated with the heat-island circulation are investigated in this paper. Similarity parameters and scaling are studied in order FIG. 1. Structure of heat-island circulation, including horizontal velocity distribution and vertical density profiles. to apply the model results to a full-scale urban heat island. Further verification of the results is recommended. 2. Theoretical formulations As discussed in the last section, a theoretical treatment is needed for low-aspect-ratio plumes. This may be accomplished by a bulk convection model or a hydrostatic model as described below. The model is formulated so that it can be applied to all low-aspect-ratio plumes, including those associated with urban heat islands, large area fires, and oceanic islands, provided that the similarity requirements are satisfied. Figure 1 shows the idealized structure of the lowaspect-ratio plume and its density distribution. Unlike high-aspect-ratio plumes, the far-field or similarity region does not appear, so that very little entrainment occurs in the upward flow. The horizontally convergent and divergent flows are adjacent and separated at the flow-reversal height z r. Therefore, the outflow rate in the upper half of the plume essentially equals the inflow rate in the lower half of the plume, which is the basis of our theoretical analysis. As shown in Fig. 1, the mixing height z i is defined as the height where the maximum difference between the plume centerline and ambient density profiles occurs. Since strong turbulent mixing occurs within the thermal plume, the potential density within the plume is nearly constant with height. The corresponding potential temperature difference between the center of the plume and the ambient surface is conventionally defined as the heat-island intensity ( m 0,or T m T m T 0 ), where m (or T m ) and 0 (or T 0 ) correspond to m and 0 in Fig. 1, respectively. The equilibrium height z e is defined as the height where the plume centerline density equals the ambient density. a. Bulk convection model A simple qualitative model for the low-aspect-ratio plume has been suggested by Briggs (1991, personal communication). The flow is assumed to be turbulent and Reynolds number independent. Only inertial and buoyancy forces are considered important, while viscous and frictional forces are neglected. We also assume that the domain is large enough that the ambient stratification T a / z at a large radial distance is not altered by the convective circulation.

3 1394 JOURNAL OF APPLIED METEOROLOGY We define ū r as the average radial inflow speed in the lower portion (to the height z r where u r 0) of the heated layer near the outer edge of the circular heat island, and w as the mean upward velocity at z r (the flow reversal height), which separates the inflow from the outflow. These definitions of ū r and w are taken somewhat loosely because the actual velocities u r and w depend on both coordinates r and z. Here, we use a cylindrical coordinate system with its origin at the center of the heat island and axis in the vertical. We use the heat-island diameter D as the length scale because it is one of the independent controlling variables that determines z i, T m, and the velocity fields. We obtain a relation for ū r assuming that the inflow near the surface is driven by the hydrostatic pressure gradient created over the center of the heat island; that is, ū r (g T m z i ) 1/2, (1) where is the coefficient of thermal expansion and g is the acceleration due to gravity. Equating the volume inflow rate to the volume upflow rate gives ū r Dz i w D 2, (2) in which we have assumed that z r z i. Also, since the upward heat flow rate is proportional to that from the surface, we obtain H 0 T m w, (3) 0 c p where H 0 is the surface heat flux, 0 is reference density, and c p is the specific heat of the fluid at constant pressure. Note that Eqs. (1) (3) essentially represent the momentum, continuity, and energy equations, all in bulk form. An additional relationship between the heat-island intensity and stratification parameter T a / z or N (g T a / z) 1/2 is obtained by assuming that z e z i,so that, from Fig. 1, g T m z i N 2. (4) Equations (1) (4) contain four unknowns, namely, ū r, w, T m, and z i, which can be solved to obtain ū r/wd const. (5) and where w /wd w D/ND, (6) z i/d w D/ND, (7) Tm wdn/g, (8) 1/3 wd (g DH 0/ 0c p) (9) is the convective velocity scale. The bulk convection model described in Eqs. (5) (9) is too simple to predict the spatial distributions of velocity and temperature; it only provides useful orderof-magnitude types of relationships between the bulk dependent variables and the controlling (independent) variables. The bulk convection model also brings out the Froude number, w D /ND, as the basic similarity parameter and w D as the fundamental velocity scale. More general similarity parameters and criteria can be obtained from the governing equations of motion and energy. The bulk model suggests the appropriate scales to be used for normalizing the governing equations: D as the length scale, w D as the radial (horizontal) velocity 2 scale, /ND as the vertical velocity scale, D/w D as the w D 2 timescale, w D N/g as the temperature scale, and 0 w D as the pressure scale. Using these scales, the normalized equations of motion and energy (see Part I) yield three nondimensional similarity parameters, namely, the Reynolds number, Re w D D/ ; the Froude number, Fr w D /ND; and the Prandtl number, Pr /, where is the kinematic viscosity and is the thermal diffusivity. The thermal plumes in the current experiments are fully turbulent, so that the Reynolds number and Prandtl number are not important (see Snyder 1981; or Part I), and the Froude number is the dominant or sole-governing parameter as predicted by the bulk convection model. b. Hydrostatic convection model Although the bulk convective model described above provides a basic framework for low-aspect-ratio plumes, it cannot provide detailed distributions of velocity or values for the proportionality constants in Eqs. (5) (8). However, if we assume that the mean state of motion is in hydrostatic equilibrium, the velocity fields can be solved analytically from the governing equations of motion. Referring again to Fig. 1, to obtain the horizontal velocity distribution and mixing height z i, we assume that the horizontal inflow velocity near the surface and the outflow velocity near the top of the plume are driven by the hydrostatic pressure difference p. This pressure difference is due to the density difference ( m a ) created by the heat island, where m is the density inside the plume and a is the ambient density outside the plume at the same elevation. We assume a steady state, so that the horizontal velocity u D (z) at the edge of the heat island (r D/2) is a function of height only. Hence, the inflow and outflow velocity u D (z)atr D/2 is given by 2 a u D /2 p. (10) This equation neglects frictional shear, which must be present near the surface and at the flow reversal height z r. Without friction, du D /dz approaches infinity at z z r and z z i, and u D 0atz 0, which are unrealistic. We also assume that the upflow is well mixed or uniformly heated, so that a constant density m is obtained inside the plume and, by definition, m matches

4 OCTOBER 1997 LU ET AL the ambient profile at z z e, the equilibrium height. The symmetry of the p profile about z e yields the boundary conditions p 0 and u D 0atz z i and z r. From the hydrostatic equilibrium assumption, p ( m )g. a (11) z It can be derived from the symmetric geometry in Fig. 1 that (see Lu 1993) z i z e z e z r,orz i 2z e z r. (12) Because m 0 z e ( a / z) and a 0 z( a / z), Eq. (11) becomes p g(z z ) a e. (13) z z Integrating this from z to z i yields p/ a. The distribution of radial velocity with height can be obtained by substituting p/ a into Eq. (10). Thus, the mixing height z i, equilibrium height z e, flow-reversal height z r, heatisland intensity T m, and volume flow rate V can be derived from the radial velocity distribution. To obtain the distribution of vertical velocity w with height, we assume that the vertical acceleration of a fluid element is due only to the density or temperature difference between the plume and the ambient environment; that is, dw g T(z), (14) dt where T(z) T m z T 0 and T a / z. At the surface where z 0, T(z) equals T m. Applying w dz/dt to Eq. (14) and integrating from 0 to z yields the vertical velocity distribution. The major results of the hydrostatic model can be summarized in nondimensional form as z i/d 1.46(w D/ND), (15) z e/d 1.013(w D/ND), (16) z r/d 0.566(w D/ND), (17) 2 2 1/2 u D/wD Z Z i at r D/2, (18) 2 1/2 w/wd 1.013(1 Z) at r 0, (19) Tm 1.013wDN/g, (20) 2 V/(D w) 0.98(w /ND), (21) D D where Z z/z e 1, Z i z i /z e 1, and V is the total volume rate of the inflow or outflow. The model based on our simplified plume geometry and Eqs. (15) (21) is referred to as the hydrostatic model; it was first suggested by Briggs (1992, personal communication). A more detailed derivation and discussion is given by Lu (1993). The model implies that the inflow velocity FIG. 2. Measured velocity fields of heat-island convection: D 44.6 cm and T/ z 0.5 C cm 1. distribution is a quarter-hyperbola, whereas the outflow and vertical velocity distributions are half-ellipses. It also suggests that the normalized upflow volume rate V/D 2 w D is a function of Froude number. Note that all the proportionality constants are determined precisely, with values depending upon the model assumptions. 3. Experimental setup As described in Part I, the convection tank used in our early experiments was that used by Willis and Deardorff in their numerous experimental studies of turbulence and diffusion in the convective boundary layer. The original tank developed some cracks after a year and was replaced by one of the same horizontal dimensions (1.20 m 1.24 m), but slightly less deep. The experimental setup was the same as that for the measurement of the temperature field, as described in Part I (see Fig. 2 in Part I). For the velocity measurements described here, only two cases were studied. They used the same stratification and heating disk but different values of heat flux. The ambient temperature gradient was 0.5 C cm 1. Changes in the temperature gradient were insignificant over the experimental runs of less than 10-min duration. The heating disk had a diameter of 44.6 cm. The heat flux values were 0.16 and 0.65 W cm 2. The mixing height z i was determined from the previous measurements of the temperature fields, which are described in Part I. Velocity fields were measured using the same tech-

5 1396 JOURNAL OF APPLIED METEOROLOGY TABLE 1. Experimental parameters for measurements of velocity fields. H 0 (W cm 2 ) No. of runs D (cm) T/ z ( C cm 1 ) w D (cm s 1 ) Fr w D /ND z i (cm) T m ( C) Re w D D/ nique as used by Willis and Deardorff (1974). Neutrally buoyant oil droplets were released into the tank as tracers. A vertical 2D plane through the center of the heater was illuminated from both sides of the tank. Multipleexposure photographs were taken with a 35-mm still camera, which was placed inside a black enclosure and faced normal to the light sheet. Both the magnitude and direction of each particle could be determined by the length and appearance of its streakline because each consisted of one longer (1-s exposure) and two shorter (0.5-s exposure) segments. Longer exposure times were used to resolve the streaklines within the low-speed region of the heat-island-induced circulation. The 2D plane was photographed in three sections to obtain sufficient resolution. Each section was photographed, and observations were repeated over several individual realizations. The average size of the oil droplets was about 200 m. The error due to the settling velocity of the particles was calculated to be about 1.5% at the typical flow speed of 1 cm s 1 at the center section; it increased to 15% at the outer fringes of the circulation region, where the vertical velocities were about 0.1 cm s 1. The streakline segments in the photographs were digitized using a microcomputer. The velocities were then averaged on 1-cm square grid cells. Scales of motion less than 1 cm were not resolved due to the minimum photographic exposure time of 0.5 s. However, this technique was able to provide an instantaneous view of the entire velocity field. The error due to the digitization process was less than 5% for the center section and around 10% for the side sections due to smaller flow speeds at the outer edges of the circulation region. More details of the oil droplet generation, dispersal, visualization, and streakline digitization system are given by Lu (1993). As discussed in Part I, the velocity fields are time dependent due to the limitation of the tank size. Only the circulation in the first few minutes of the quasisteady state are considered to be representative of the full-scale case. Hence, all the velocity measurements were conducted during the initial phase of the quasisteady state. 4. Experimental results and discussion a. Mean velocity vectors and streamlines The velocity fields associated with the axisymmetric circulation were measured in two cases using the same heater size and temperature stratification but different FIG. 3. Streamline contours. Solid lines: counterclockwise motion. Dashed lines: clockwise motion. (a) H W cm 2 and (b) H W cm 2. heat fluxes as shown in Table 1. Each case was repeated several times. The velocity field for each realization was measured from the streakline photographs. Velocity vectors were averaged for each 1-cm square grid cell over 110 cm 12 cm and 110 cm 18 cm domains for the low and high heating rate cases, respectively (heating rates were 0.16 and 0.65 W cm 2 ). Each grid cell contained approximately 20 velocity vectors, whose averages are shown in Fig. 2. From these figures one can clearly see the strong updrafts at the center of the heat island, compensating downdrafts aloft around the periphery, converging horizontal motions in the lower half of the plume, and diverging motions in the upper half. A more clear depiction of the circulation for the two cases is given in Fig. 3 in the form of mean streamline contours. These were calculated from the velocity fields shown in Fig. 2 using the relations d rwdr rurdz, (22) 1 1 ur, and w, (23) r z r r where (r, z) is the streamfunction for a 2D flow in a cylindrical coordinate system. Streamlines are contours of constant, with positive values (shown as solid lines in Fig. 3) representing counterclockwise circulation and negative values (dashed lines) indicating clockwise motion. The streamline patterns for the two cases with different Froude and Reynolds numbers are quite similar. The inward flow near the surface and the outward flow aloft at the periphery of the dome are not strictly horizontal because the divergent flow at the equilibrium

6 OCTOBER 1997 LU ET AL FIG. 4. Nondimensional radial velocities at different heights for (a) H W cm 2 and (b) H Wcm 2. FIG. 5. Nondimensional radial velocities at different locations for (a) H W cm 2 and (b) H W cm 2. height cannot spread out very far before subsiding due to the presence of the sidewalls. b. Radial velocity Figure 4 shows the normalized radial velocity u r /w D as a function of normalized distance and height. The distributions for the two cases are quite similar. Near the surface, the magnitude of the radial inflow velocity increases going toward the center, reaching a maximum around r/d 0.25, then decreases to zero at the center of the heat island. The location of the maximum in u r depends upon height. The outflow speed aloft also increases away from the center to a maximum near r/d 0.25, then decreases with further increases in radial distance. It is interesting to note that outside the periphery of the heat island, the magnitude of the radial velocity is inversely proportional to the radial distance. This type of behavior is predicted by the continuity equation; if one neglects the mean vertical motion in the far-field region outside the heat island, u r /r u r / r 0, (24) which may be integrated to yield u r (z) r 0 u 0 (z)/r, (25) where u 0 (z) is the radial velocity at a particular location r 0 D/2, which may depend upon height. Both the bulk-convection and hydrostatic models discussed earlier predict that the average inflow velocity near the edge of the heat island should be scaled by w D. The similarity of the normalized radial velocity distributions for the two cases indicates that w D may be the appropriate velocity scale for the entire radial velocity field, that is, u r /w D is a function of r/d and z/z i only. Whether the distribution also depends upon Froude number cannot be determined on the basis of only two cases. The normalized vertical profiles of radial velocity for the two cases are shown in Fig. 5. The velocities at the surface must be zero at all radial distances. The approach to zero velocity occurs within the thin viscous sublayer. The flow field very close to the surface (z/z i 0.03), including the viscous sublayer, could not be determined by the measurement technique used in our experiments. The measured profiles of radial velocity imply the existence of sharp maxima and very large velocity gradients within the viscous sublayer close to the surface. But such a viscous sublayer is not expected to apply to the urban-heat-island-induced circulation over an aerodynamically rough surface comprising a thick urban canopy layer.

7 1398 JOURNAL OF APPLIED METEOROLOGY FIG. 6. Nondimensional vertical velocities at different locations for (a) H W cm 2 and (b) H Wcm 2. The radial velocity profile near the edge of the heat island (r/d 0.5) as deduced from the hydrostatic model [Eq. (18)] is also plotted in Fig. 5 for comparison with the measurements. The hydrostatic model provides a reasonable prediction of the shape of the profile except near z 0 and z z r, due to the assumption of frictionless flow. However, Eq. (18) overestimates the magnitudes of the radial velocities by about a factor of 2 in comparison with the experimental results. This is due to the crude assumptions of (a) uniform mixing inside the plume and (b) the absence of turbulent friction in the hydrostatic model; both assumptions enhance the flow speeds. c. Vertical velocity Figure 6 shows the observed profiles of normalized vertical velocity with scaling based upon the bulk convection model. The profile near the center (r/d 0.01) shows the maximum velocity occurring in the middle of the plume. The maximum velocity as well as the height of the maximum decreases with increasing radial distance from the center, and the downward motion aloft becomes more prominent. A comparison of the normalized profiles for the two cases shows only an approximate similarity, indicating possible dependence of FIG. 7. Nondimensional vertical velocity w/w D Fr distributions above heat island: (a) H W cm 2 and (b) H W cm 2. the velocity field on the Froude number. The scaling of vertical velocity by w D alone (see Fig. 5), as suggested by the hydrostatic model [Eq. (19)], was not found to be as effective as scaling by w D Fr (see Fig. 6) for the two cases studied. The vertical velocity distributions at different heights across the plume for the two cases are shown in Fig. 7. The width of the upflow region is nearly constant from z/z i 0.2 to 0.6. A stationary wave is evident at the

8 OCTOBER 1997 LU ET AL u maxin /w D r/d z/z i u maxout /w D r/d z/z i w max /w D r/d z/z i TABLE 2. Locations of the largest velocities and flow-reversal height. H W cm 2 H Wcm 2 Average z r /z i equilibrium height (z/z i 0.7) in both cases. The high heating rate case exhibits a larger amplitude due to the larger Froude number. This is a consequence of the overshooting of the thermal plume above its equilibrium height in the stably stratified environment. The plume shape or boundaries could be defined in several ways: as an instantaneous visible boundary from photographs or as the time-averaged boundaries determined from velocity or temperature measurements. Whereas different definitions may not give exactly the same boundary, they should give approximately similar shapes. The minimum plume width l min, determined by the upflow region in the middle of the plume in Fig. 7 is about half the heat-island size D that is, the contraction ratio l min /D 0.5. Different values of this contraction ratio have been reported by other investigators on the basis of similar experiments. For instance, Stout (1986) showed that the diameter of the plume boundaries shrank to 43% of the original diameter based on density profile measurements. Palmer (1981) stated that in Project Flambeau the convection column shrank to about two-thirds of the diameter of the fire at the base. Faust (1981) also conducted experiments above a circular hot-plate in water and photographed a contraction to 20% of the original diameter using a Schlieren system. Husar and Sparrow (1968) have shown a contraction to one-half the original diameter for a plume originating from a circular hot-plate in water. The different values of the contraction ratio from different experiments suggest that it may depend upon experimental conditions. For a low-aspect-ratio plume, the larger the aspect ratio of the plume, the smaller the contraction ratio. Therefore, a small contraction ratio indicates a large Froude number, as suggested by Eq. (15). The minimum width of the plume due to flow contraction can also be estimated from the hydrostatic model [e.g., Eq. (21)]. Due to continuity, the minimum plume width l min must be associated with the maximum upflow speed w max, which appears at z e. With z z e, Eq. (19) gives w max 1.01w D. (26) The upflow volume rate at z e equals the inflow volume FIG. 8. Locations of maximum inflow, maximum outflow, maximum upflow, and flow-reversal height for H W cm 2 and H W cm 2. rate V in Eq. (21). Therefore, from the above relation and Eq. (21), we obtain l min 1.11(w D /ND) 1/2 D. (27) The above inequality provides only the lower bound for the plume width at z e because the average upflow speed at z e is smaller than the centerline value w max ( 1.01 w D ). It also suggests that l min /D is a function of Froude number for the low-aspect-ratio thermal plume. The two contraction ratios observed from Figs. 7a and 7b are about the same at 0.5. The calculated values of 1.11(w D / ND) 1/2 are 0.24 and 0.31 for the low- and high-heatingrate cases, respectively. Although our experimental data are not adequate to verify this prediction, Eq. (27) gives at least a partial explanation for the variations of l min /D in different studies and provides direction for further investigation. d. Maximum velocities, flow-reversal height, and various height ratios The locations and values of the maximum radial inflow velocities u maxin /w D, the maximum outflow velocities u maxout /w D, and the maximum upflow velocities w max / w D are listed in Table 2. The normalized magnitudes of the maxima in both cases are approximately the same ( 0.5). The radial locations of the maximum inflow and outflow speeds (r/d 0.25) are fairly close to each other. The locations of the local maximum inflow, outflow, and upflow velocities, as well as the flow-reversal (zero flow) height (z r /z i ) for the two cases, are shown in Fig. 8. The good agreement between the two cases suggests that w D, D, and z i are proper scaling parameters for the maximum velocities. The maximum radial velocities and their locations, based on the two cases, do not depend significantly on Froude number. Further studies are needed with wider ranges of w D and Fr to confirm Froude number independence (or ascertain a weak dependence). The measured zero-velocity (flow reversal) heights as

9 1400 JOURNAL OF APPLIED METEOROLOGY TABLE 3. Ratios between z i, z e, and z r for both laboratory observations and theoretical predictions. Ratios z e /z i z r /z i z r /z e Laboratory observation Hydrostatic prediction functions of r/d are also plotted in Fig. 8. The mean value of z r /z i outside the central plume region (r/d 0.25) is about The hydrostatic model [Eqs. (15) and (17)] predicts that z r /z i 0.387, which is very close to the experimental result. Therefore, z r /z i may also be used as a predictor to determine the flow-reversal height for low-aspect-ratio thermal plumes. The two observations of the flow-reversal height z r /D suggest that z r /D 1.03 Fr. (28) The hydrostatic prediction [Eq. (17)] is about 45% lower than that of Eq. (28) based on observations. This may be due to the idealized geometry of the hydrostatic model, which does not account for overshooting of a lowaspect-ratio plume, and the assumed uniform mixing inside the plume. The hydrostatic predictions of z i /D and z e /D [Eqs. (15) and (16)] are 49% and 44% lower than the measured values, respectively (see Part I). However, the ratios z e /z i, z r /z i, and z r /z e from both the laboratory measurements and the hydrostatic model predictions are very close to each other. These ratios are listed in Table 3, in which z i and z e are determined from the temperature measurements in Part I. The good agreement suggests that the hydrostatic model is able to accurately predict the various height ratios, but not the absolute heights; it underestimates the heights z e, z r, and z i by 44% to 49%. e. Standard deviations of velocities Standard deviations of radial and vertical velocities in each grid cell were calculated by subtracting their mean values from some 40 instantaneous values obtained from the streakline photographs (the calculations took advantage of the symmetry about the center). Normalized standard deviations u /w D and w /w D at the center of the heat island are plotted as functions of z/z i in Fig. 9. Also shown in the figure are some nighttime field data from the city center of Sapporo, Japan (Uno et al. 1988, 1992). (The values of z i and w D in the field were determined in the same way as described in section 2 for the tank, using full-scale temperature profiles to determine z i and published values for the Sapporo city size D and surface heat flux H 0.) The wind speeds for both runs 15 and 17 were calm near the surface ( 1 ms 1 ) and light aloft to the inversion height ( 3 ms 1 ). At the top of the inversion (44 m for run 15 and 59 m for run 17), there were small jets (4.6 m s 1 for both cases). Therefore, the field data may be considered to have light wind conditions. Both laboratory and field data show larger values of u / w D and w /w D within the mixed layer and a rapid decrease with height above z i. The maximum occurs near the top of the mixed layer in each case. The approximate similarity between the laboratory and field data should be noted, although the tank values of u /w D are on the low side and those of w /w D are on the high side of the field values (which show very large scatter). The differences in turbulence intensities may be attributed to experimental errors. The confinement of the heat-island-induced circulation within a finite-sized tank probably suppresses horizontal motions and thus the tank values of u /w D may be underestimated. Deardorff and Willis (1985) noticed the same effects in their simulation of the convective boundary layer in the same tank. The contour plots of u /w D and w /w D for the entire FIG. 9. Nondimensional standard deviations of velocity vs z/z i at the centerline above the heat island for both laboratory and field data: (a) horizontal velocity and (b) vertical velocity.

10 OCTOBER 1997 LU ET AL flow field (see Lu 1993) show a region of high turbulence along the top of the thermal plume. The region of maximum u /w D extends farther horizontally than that of w /w D due to horizontally divergent flow caused by stable stratification at the plume top. This divergent flow creates larger fluctuations in the horizontal component than in the vertical component of motion. 5. Concluding remarks The behavior of a low-aspect-ratio plume is fundamentally different from that of a high-aspect-ratio plume. The high-aspect-ratio plume grows with height due to entrainment of ambient fluid into the axial flow, whereas there is little entrainment from the ambient in the low-aspect-ratio plume. Therefore, flow features associated with the low-aspect-ratio plume, such as the radial and vertical velocity distributions, are completely different from those associated with the high-aspectratio plume. For this reason, classical plume theories, which were developed for high-aspect-ratio plumes, cannot be applied to low-aspect-ratio plumes. We have formulated two simple analytical models of the lowaspect-ratio plume and compared the model predictions with the results of our laboratory experiments. The velocity scale w D (g DH 0 / 0 c p ) 1/3 is suggested by both the bulk convection model and the hydrostatic model as a scaling parameter for both radial and vertical velocities. Two cases with different values of heat flux and, hence, different values of w D, were studied experimentally to test the validity and effectiveness of this scaling. It was found that w D could scale the entire radial velocity field satisfactorily on the basis of the two cases studied. Uncertainties remain for the vertical velocity field as the Froude number is involved, and the scaling by w D alone was found not to be as effective as scaling by w D Fr for the two cases studied. The normalized values and locations of the maximum inflow, outflow, and upflow speeds for the two cases agreed very well. These values are essentially constants and the ratio between the plume heights z e /z i, z r /z i, and z r /z e are also constants in good agreement with the predictions of the hydrostatic model. The parameter w D appears to scale the standard deviations of radial and vertical velocities u and w fairly well and agrees with the field observations favorably, even with large differences in the Reynolds numbers and Froude numbers between the laboratory and field experiments. The flows produced in our laboratory model are essentially Reynolds number independent, and the plumes are in the low-aspect-ratio category. Therefore, the velocity scale w D and the experimental results may be applied to other low-aspect-ratio turbulent plumes, such as urban heat islands. Since the technique used for velocity measurements in our experiments was very time consuming, only two cases were studied; thus, no definite conclusions can be drawn on the possible dependence of normalized velocity field on Froude number. Further investigations including numerical studies are needed. In fact, urban areas are generally more polluted than their rural environs, and the pollutants are usually confined within the urban plume. Therefore, the urban plume height is one of the most important factors in urban air pollution assessment. Our results show that the plume height is a function of Froude number only, which could be used in urban air quality models to estimate the impact of urban pollution. Although the scaling parameters Fr, W D, and D found in this study may be useful in urban air quality models, a more systematic study is desired. Acknowledgments. We wish to acknowledge the valuable comments and suggestions made by Dr. Gary Briggs, NOAA. His contributions to the development of the theoretical models and similarity scaling are especially recognized. Technical support provided by the entire staff of the EPA Fluid Modeling Facility is gratefully acknowledged. Disclaimer. The information in this document has been funded wholly or in part by the U.S. Environmental Protection Agency under Cooperative Agreement CR with the North Carolina State University. It has been subjected to Agency review and approved for publication. Mention of trade names or commercial products does not constitute endorsement or recommendation for use. REFERENCES Briggs, G. A., 1969: Plume Rise. Critical Review Series, U.S. Atomic Energy Commission, 81 pp. [NTIS TID ], 1975: Plume rise predictions. Lectures on Air Pollution and Environmental Impact Analysis, Amer. Meteor. Soc., Deardorff, J. W., and G. E. Willis, 1985: Further results from a laboratory model of the convective planetary boundary layer. Bound.-Layer Meteor., 32, Faust, K. 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