Plume Width Research Articles

A laboratory study of buoyant plumes in laminar and turbulent crossflows

The results of laboratory experiments of buoyant plumes in crossflows are presented. Specific comparisons are made of plume evolution for laminar and weakly turbulent crossflows for otherwise identical flow parameters to establish that plume dilution rates differ significantly even for weakly turbulent crossflows. Decaying, grid-generated turbulence constituted the flow field of the weakly turbulent crossflow. Measurements comprised analyses of flow visualization, hot-film anemometry and conductivity probe data. Plume thicknesses and widths are found to be greater for turbulent crossflows, although they both grow as χ 2/3 for laminar and turbulent crossflows. The effect of turbulence in the crossflow is to increase entrainment and dilution with the consequence of lower plume trajectories. It is found, relative to the values of dilution for buoyant plumes in a laminar crossflow, that dilutions of plumes in a weakly turbulent crossflow (with kinetic energy dissipation rates two orders of magnitude smaller than internal plume dissipation rates) were greater by 33%. Trajectories can be predicted by an integral model with a modified β, the entrainment coefficient, which relates the ratio of dissipation rates within and outside the plume.

Near-field mixing at an outfall

The near-field mixing of effluent from buoyant and nonbuoyant flows from outfalls discharging into a cross-flowing ambient current in trapezoidal channels has been investigated. A physical model was scaled to represent a typical large outfall into one of the connecting channels of the Great Lakes system. The discharged jet was measured in detail to determine the velocity and concentration fields. The excess velocities and concentrations were found to follow the Gaussian distribution. Empirical expressions for the jet trajectories, minimum dilutions, and plume widths were derived. A correction factor has been introduced to the concentration distribution to ensure mass conservation. The calibrated model was verified by comparison with field data and the results of other near-field models (PDS, MIT, and CORMIX). Key words: mixing, near field, pollutant transport, outfall, buoyant flow, non-buoyant flows.

Environmental coating problems

A method for the determination of atmospheric heavy metal deposition rates has been developed using 0.5-m2 deposition trays at 0.1 m from the ground. Trays were spaced at 150-m intervals along a 1500-m line 500 m east of a Pb–Zn smelter. Ten sampling events of 1–3-h duration were conducted under westerly wind conditions so as to determine the sources of heavy metals deposited near the smelter. Deposited materials were sampled from the trays using wipes. There was good agreement between deposition trays placed side-by-side and exposed in pairs. Under certain conditions, however, the method is not appropriate owing to the potential for local contamination. Geometric mean deposition rates for Pb, Zn, Fe, Cu, As and Cd averaged over a nominal plume width of 600 m amounted to 18.8, 22.2, 12.2, 0.614, 0.403 and 0.052 mg m−2 day−1, respectively. Gaussian deposition profiles were seen for Pb, Zn, Fe, Cu, As and Cd downwind from the blast furnace, sinter plant, and refinery area. Zinc deposition could also be attributed to a northern Zn production area. This northern site was not generally associated with elevated Pb deposition. On the basis of this work, the deposition of heavy metals in residential areas adjoining the smelter is likely to occur downwind from the smelter site, with deposition rates increasing with wind speed. The strategic measurement of heavy metal dry-deposition rates over short periods of time using large collection surfaces provides source-specific information not obtainable by conventional long-term ‘passive’ deposition sampling. Lower detection limits than those achieved here are likely to be achieved in non-smelter settings. Previous suggestions implicating a sink of city surface dusts as the probable source of Pb recontamination of residential settings in the absence of ongoing smelter emissions are not supported by this work.

Estimate of diffusion parameters of aircraft exhaust plumes near the tropopause from nitric oxide and turbulence measurements

Horizontal and vertical plume scales and respective diffusivities for dispersion of exhaust plumes from airliners at cruising altitudes are determined from nitric oxide (NO) and turbulence data measured with the DLR Falcon research aircraft flying through the plumes. Ten plumes of known source aircraft were encountered about 5 to 100 min after emission at about 9.4 to 11.3 km altitude near the tropopause in the North Atlantic flight corridor at 8°W on three days in October 1993. The ambient atmosphere was stably stratified with bulk Richardson numbers greater than 10. The measured NO peaks had half widths of 500 to 2000 m with maximum concentrations up to 2.4 parts per billion by volume (ppbv), clearly exceeding the background values between 0.13 and 0.5 ppbv. For analysis the measured plumes are approximated by an analytical Gaussian plume model which accounts for anisotropic diffusion in the stably stratified atmosphere and for shear. Two methods are given to obtain diffusivity parameters from either the individual plume data or the set of all plume measurements. Using estimates of the emitted mass of NO per unit length, the vertical plume width is found to be 140 m on average. This width is related to mixing in the initial trailing vortex pair of the aircraft. The range of the plume data suggests vertical diffusivity values between 0 and 0.6 m2 s−1. The turbulence data exhibit strong anisotropic air motions with practically zero turbulent dissipation and weak vertical velocity fluctuations. This implies very small vertical diffusivities. The horizontal diffusivity is estimated as between 5 and 20 m2 s−1 from the increase of horizontal plume scales with time. For constant diffusivities, shear dominates the lateral dispersion after a time of about 1 hour even for the cases with only a weak mean shear of 0.002 s−1.

Characterization of mantle convection experiments using two‐point correlation functions

Snapshots of the temperature T(r,ϕ,t), horizontal flow velocity u(r,ϕ,t), and radial flow velocity w(r, ϕ, t) obtained from numerical convection experiments of time‐dependent flows in annular cylindrical geometry are taken to be samples of stationary, rotationally invariant random fields. For such a field ƒ(r,r2 ϕ, t), the spatio‐temporal two‐point correlation function, Cƒƒ (r, r1, Δ, t*), is constructed by averaging over rotational transformations of this ensemble. To assess the structural differences among mantle convection experiments we construct three spatial subfunctions of Cƒƒ (r, r1, Δ, t*) : the rms variation, σƒ (r), the radial correlation function, Rƒ (r,r1), and the angular correlation function, Aƒ(r, Δ). Rƒ (r,r1) and Aƒ(r,Δ) are symmetric about the loci r = r1 and Δ=0, respectively, where they achieve their maximum value of unity. The falloff of Rƒ and Aƒ away from their symmetry axes can be quantified by a correlation length ρƒ(r) and a correlation angle αƒ(r), which we define to be the half widths of the central peaks at the correlation level 0.75. The behavior of ρƒ is a diagnostic of radial structure, while αƒ measures average plume width. We have used two‐point correlation functions of the temperature field (T‐diagnostics) and flow velocity fields (V‐diagnostics) to quantify some important aspects of mantle convection experiments. We explore the dependence of different correlation diagnostics on Rayleigh number, internal heating rate, and depth‐ and temperature‐dependent viscosity. For isoviscous flows in an annulus, we show how radial averages of σT, ρT, and αT scale with Rayleigh number for various internal heating rates. A break in the power‐law relationship at the transition from steady to time‐dependent regimes is evident for ρT and αT but not for σT or the Nusselt number. A rapid tenfold to thirtyfold viscosity increase with depth yields weakly stratified flows, quantified by aw, which is a measure of radial flux. The horizontal flux diagnostic, σu, reveals that the flow organization is sensitive to the depth of the viscosity increase. A jump at middepth induces a significant horizontal return flow at the base of the upper layer, absent in models with a jump at quarter‐depth. We illustrate that T‐diagnostics, which are more easily relatable to geophysical observables, can serve as proxies for the V‐diagnostics. A viscosity increase with depth is evident as an increase in the T‐diagnostics in the high‐viscosity region. For numerical experiments with a temperature‐dependent rheology we employ a mobilization scheme for the upper boundary layer. Temperature dependence does not appreciably perturb the σ‐diagnostics or αT in the convecting interior. Changes in the radial correlation length are twofold. First, the greater viscosity of cold downwellings leads to an increase in height and width of the radial correlation maximum near the top. Second, the increase in ρT associated with a viscosity jump is markedly reduced. The latter effect can be explained by weaker, less stationary hot upwellings, mobilized by the temperature‐dependent rheology and disrupted by the cold, high‐viscosity downwellings.

The Amazon River Plume during AMASSEDS: Spatial characteristics and salinity variability

The Amazon River discharge forms a plume of low‐salinity water that extends offshore and northwestward over the north Brazilian shelf. Observations acquired as part of A Multidisciplinary Amazon Shelf SEDiment Study (AMASSEDS) are used to characterize the spatial structure and temporal variability of the Amazon Plume. Four shipboard conductivity‐temperature‐depth (CTD) surveys spanning the shelf from 1°S to 5°N during rising (March 1990), maximum (May 1990), falling (August 1989), and minimum (November 1991) discharge show the Amazon Plume is typically 3 to 10 m thick and 80 to over 200 km wide. Northwest of the river mouth, the plume is often characterized by a wedge of low‐salinity water adjacent to the coast and a separate tongue of low‐salinity water extending offshore over the middle to outer shelf. A bottom front separating the low‐salinity plume water from oceanic water is consistently located between the 10‐ and 20‐m isobaths. A moored array deployed about 300 km northwest of the river mouth from February to June 1990 included inner and midshelf moorings in 18 and 65 m of water on which temperature‐conductivity measurements were made. The moored observations reveal salinity variations within the Amazon Plume of over 10 psu on timescales of days to weeks. This variability includes intermittent events in which plume water pools up in the vicinity of the river mouth and the plume width can exceed 200 km. These accumulation events are apparently due to wind events with a southeastward component which impede or block the normally northwestward freshwater transport. The resulting bulges in the plume are then released northwestward when the wind reverses. Volume budgets indicate the Amazon Plume entrains roughly twice the river discharge between the river mouth at the equator and 3°N. Estimates of gradient Richardson numbers from the moorings suggest entrainment, due to the strong semidiurnal tidal currents, occurs where the plume intersects the bottom and over the outer portion of the plume, where salinities approach oceanic values.

The fluid dynamics of plume-ridge and plume-plate interactions: An experimental investigation

A number of hotspots occur at or near mid-ocean ridges (e.g., Iceland) causing elevated bathymetry, gravity anomalies, and geochemical enrichment for hundreds of kilometers along otherwise normal ridge axes. This is commonly presumed to be the result of hot plume material from the deep mantle flowing outward from a narrow plume conduit. Here we explore the dynmics of plume-ridge and plume-plate interaction using laboratory experiments. We model chemically buoyant plumes, as opposed to thermally buoyant plumes, so we ignore the effects of thermal diffusion while concentrating on the strictly fluid dynamical aspects of the interaction. We find that the spreading of a continuously fed plume beneath a fixed plate is described by a simple expression derived from a balance between plume buoyancy and viscous resistance to spreading. For plumes spreading beneath ridges we find two remarkably distinct phases of flow: At first the plume spreads approximately as it would beneath a fixed plate, then a rapid transition occurs to a steady-state condition in which the plume width along the ridge (‘waist’) is constant. This width is found to be proportional to a natural length scale that increases with the square root of the plume volumetric flux and the ratio of the mantle/plume viscosities, and decreases with the square root of the plate spreading velocity. Although our experimental conditions, with uniform mantle viscosity and no thermal diffusion, are perhaps oversimplified compared to the real Earth, the results scale reasonably well to observation. Our model suggests that sufficient time has elapsed (∼ 60 Myr) for the Iceland plume to reach steady state in its interaction with the Mid-Atlantic Ridge. Using independent estimates for the plume volumetric flux at the Iceland hotspot, and assuming a viscosity ratio of one, the model predicts a minimum waist width of about 850 km, in close agreement with the observed value of about 920 km. Data from other hotspots (Jan Mayen, Azores, Galapagos) are also consistent with the experimental results for average mantle/plume viscosity ratios of 1–3. Using estimated plume fluxes with a simple conservation of mass equation yields reasonable estimates of plume thickness ranging from 70 to 180 km for these hotspots. Thus the experimental model appears to be reasonably consistent with observation, given that thermal diffusion has been disregarded.

Buoyant Plume through a Permeable Porous Layer Located above a Line Heat Source in an Infinite Fluid Space.

An experimental and analytical study is conducted on the effects of a horizontal porous layer on the development of the buoyant plume arising from a line heat source in an infinite fluid space. Visual observations and the experimental temperature distributions show the expansion and contraction of the plume at the lower and upper interfaces of the permeable layer. Similarity solutions assuming their proper virtual origins can approximately predict the changes of the plume width. Detailed numerical examination indicates that the lateral flow induced near the interfaces causes deviation from the similarity solutions. The vicinity of the interface is understood to be a transition region between the similarity solutions. The numerical analysis adopts the Beavers-Joseph slip boundary condition with the slip coefficient α estimated by α=1/√(e), where e is the porosity of the porous layer. The fairly good agreement between the experimental and numerical results confirms the validity of the slip coefficient.

Effluent from diffuse hydrothermal venting: 2. Measurement of plumes from diffuse hydrothermal vents at the southern Juan de Fuca Ridge

An array of temperature and velocity sensors was deployed for two weeks in the bottom boundary layer at the Southern Juan de Fuca Ridge during the National Oceanic and Atmospheric Administration VENTS'90 field experiment. The instruments were placed close to the seafloor to detect effluent from diffuse hydrothermal sources located 1–2 km distant. Over the duration of the field deployment, we detected effluent from two vent sources. The plume from the larger of the two sources was sensed at all of the instrument locations, permitting estimates of dilution based on the field measurements. Our data also permitted estimation of energy flux from diffuse vents (approximately 125±75 MW from a source to the north of the instrument array, and 12±8 MW from a small source to the east). The largest source of error in the data was estimation of the plume width. We have described a sensor array that would reduce this error in future work. Such an array would be suitable for a long‐term vent activity monitor.

Measurements of bubble plume behaviour and flow velocity in gas stirred liquid Wood's metal with an eccentric nozzle position

The distribution of gas fraction and the flow field of gas‐stirred liquid metal in steel ladles at eccentric injection of the stirring gas through the bottom of the vessel were measured in melts of 437 kg liquid Wood's metal. The melts had a temperature of 100°C. The bath height was 37 cm and the vessel diameter 40 cm. The blowing nozzle was positioned at half of the vessel radius. Gas flow rates were between 100 and 800 cm3(STP)/s. The gas fractions were measured by electrical resistance probes. The flow velocity of the liquid metal was determined by magnet‐probes. The gas fraction and the velocity distribution in the plume were found to have a Gaussian shape. The cross‐section of the plume is ellipsoid, as the plume width in the direction of the radius was a little smaller than the width in the direction perpendicular to it. Moreover the plume was inclined to the wall. The results which were found for the plume are mathematically described. The flow field at eccentric gas‐stirring consists of one great loop, which fills almost the entire vessel. This is contrary to centric blowing, where for aspect ratios of the ladle in the order of 1, a toroid is formed in the upper and a dead zone exists in the lower part of the vessel. The consequences of this behaviour, especially for mixing in the melt, are discussed.

Wind-tunnel studies of roughness effects in gas dispersion

Abstract A programme of work has been carried out using wind-tunnel modelling to investigate the effect of surface roughness on dense-gas dispersion from ground-level sources where the roughness height is large compared with the cloud depth. Neutrally buoyant gas releases were included as control experiments. The results are compared with results of a 1:100 scale simulation of dispersion in high roughness conducted by CPP Inc. for the American Petroleum Institute (API). For surface roughness lengths, z0, smaller than about 0.5 cm (0.5 m full scale), dispersion is shown to be well described by a Gaussian plume model. Plume width and depth increase with z0 and are simple functions of distance from the source. For larger roughness, dispersion depends on the arrangement of the roughness elements, plume width, σy, can be restricted by channelling and plume depth, σz, becomes sensitive to run conditions.

Injection of bubbling liquid jets from multiple injectors into a supersonic stream

Multiple (12 injectors) bubbling liquid jets (helium microbubbles in water) were injected transverse to a M = 2.4 airflow. Penetration and spray plume spreading angle were measured directly using nanoshadowgraphs and front-lighted pictures, respectively. The experiments were performed at two conditions, i.e., the constant supply pressure condition and the constant liquid mass flow rate condition. For the case of a parallel arrangement of the injector orifices to the airflow, the penetration of the jet array increased steadily from front to back. The last jet (12th jet) has over 5 times the penetration of the first jet for the water only case. The usual similarity law for the penetration, h * q°-59 was approximately valid also for the multiple water-only jets. For the bubbling jet case, the penetration of the first jet doesn't change with increasing gas concentration y, but the rear jets have less penetration height than that of liquid-only jets at the constant injection pressure condition. For the constant injection pressure condition, the resulting penetration of the jet plume decreased with increasing y. On the other hand, for the constant liquid mass flow rate condition, the penetration of the multiple bubbling jets increased a little with increasing y. Straight coherent jets just coming out of orifice were observed for the y = 0 case. Conical jet plumes were obtained for the bubbling jet case. Therefore, the width of the jet plume increased by using the bubbling jet. The effects of the angle between the orifice array and the freestream direction and the surfactant concentration on the penetration and mixing of multiple bubbling jets were also clarified.

Two-Zone Convective Scaling of Diffusion in Complex Terrain

A “two-zone” convective scaling equation is developed that allows for a step change in scaling parameters. The results of the Lompoc Valley Diffusion Experiment, conducted at coastal site in California under variable cloud cover, are compared to the two-zone model and five other parameterizations obtained from the literature. The two-zone method is shown to unambiguously select appropriate convective scaling parameters for the partially cloud-covered cases. Its accuracy is comparable to a statistical approach that relies on in situ turbulence measurements and a Lagrangian time scale derived from the data. While the two-zone method works for maximum concentration predictions at any downwind distance, all models underpredict plume width and crosswind-integrated concentration measured at long ranges. It is speculated that the observed enhancement is due to near-surface topographic flows that spread the plume at the surface but maintain centerline concentration.

Plumes Above Line Fires in a Cross-Wind

A model for the plume above a line fire in a cross wind is constructed. This problem is shown to reduce to numerically solving a system of 6 coupled ordinary differential equations for given initial conditions that depend upon the fire characteristics. The model is valid above the flaming zone and takes inputs such as the width, velocity and temperature of the plume at a given height above the flaming zone, Different horizontal ambient wind velocities are allowed for and a comparison is made between some of these representative wind profiles. The plume trajectory, width, velocity and temperature are calculated for these different representative velocity profiles. This model has application to the calculation of temperature-time exposures of vegetation above line fires and hence can be used in models that predict effects such as leaf scorch and canopy stored seed death. On a larger scale it has application to the problem of tracking burning brands which can cause spotting ahead of the fire.

Hazardous gas model evaluation with field observations

Fifteen hazardous gas models were evaluated using data from eight field experiments. The models include seven publicly available models (AFTOX, DEGADIS, HEGADAS, HGSYSTEM, INPUFF, OB/DG and SLAB), six proprietary models (AIRTOX, CHARM, FOCUS, GASTAR, PHAST and TRACE), and two “benchmark” analytical models (the Gaussian Plume Model and the analytical approximations to the Britter and McQuaid Workbook nomograms). The field data were divided into three groups—continuous dense gas releases (Burro LNG, Coyote LNG, Desert Tortoise NH 3-gas and aerosols, Goldfish HF-gas and aerosols, and Maplin Sands LNG), continuous passive gas releases (Prairie Grass and Hanford), and instantaneous dense gas releases (Thorney Island freon). The dense gas models that produced the most consistent predictions of plume centerline concentrations across the dense gas data sets are the Britter and McQuaid, CHARM, GASTAR, HEGADAS, HGSYSTEM, PHAST, SLAB and TRACE models, with relative mean biases of about ±30% or less and magnitudes of relative scatter that are about equal to the mean. The dense gas models tended to overpredict the plume widths and underpredict the plume depths by about a factor of two. All models except GASTAR, TRACE, and the area source version of DEGADIS perform fairly well with the continuous passive gas data sets. Some sensitivity studies were also carried out. It was found that three of the more widely used publicly-available dense gas models (DEGADIS, HGSYSTEM and SLAB) predicted increases in concentration of about 70% as roughness length decreased by an order of magnitude for the Desert Tortoise and Goldfish field studies. It was also found that none of the dense gas models that were considered came close to simulating the observed factor of two increase in peak concentrations as averaging time decreased from several minutes to 1 s. Because of their assumption that a concentrated dense gas core existed that was unaffected by variations in averaging time, the dense gas models predicted, at most, a 20% increase in concentrations for this variation in averaging time.

Dispersion and deposition of smoke plumes generated in massive fires

Massive fires resulting from the uncontrolled burning of crude oil from spills or industrial accidents produce large smoke-laden buoyant plumes which rise in the wind direction before they equilibrate within a stably stratified atmosphere. Beyond this point, the plume material cools by entrainment and the plume becomes negatively buoyant due to the heavy smoke loading. The trajectory of the descending plume, which determines the ground distribution of smoke, is the subject of this paper. A computational model for the simulation of large-scale smoke plumes resulting from such fires is developed and applied to investigate the effects of the plume initial properties on its trajectory and smoke deposition patterns. Attention is focused on the descent and dispersion of wind-driven plumes in a homogeneous atmosphere, and the smoke deposition on flat terrain. Results show that the plume dynamics in the cross-wind direction are dominated by two buoyantly generated, coherent, streamwise vortices which distort the plume cross section into a kidney-shaped structure. The strength of the two vortices and their separation increase as the plume falls. The plume width grows under the action of these vortices at a rate which increases as the plume settles on the ground, leading to a smoke footprint which does not resemble the prediction of Gaussian dispersion models. The effects of the injection altitude and the initial shape of the plume cross section on the transport and dispersion of the negatively buoyant smoke plume are investigated. Plumes falling from higher elevations disperse more in the vertical direction while those falling from lower elevations disperse further in the horizontal cross-wind direction. Plumes with circular cross-sections reach the ground faster and disperse horizontally further than plumes with elliptical cross-sections with the minor axes in the vertical direction. Vertical plume dispersion is weakly dependent on the shape of its initial cross-section.

Buoyant Plume through a Permeable Porous Layer Located above a Line Heat Source in an Infinite Fluid Space.

An experimental and analytical study is made on the effects of a horizontal porous layer on the development of the buoyant plume arising from a line heat source. In the experiments, the porous layer is simulated by an array of acrylic resin plates spaced with a small gap width according to the Hele-Shaw analogy. Visual observations and experimental temperature distributions show the expansion and contraction of the plume at the lower and upper interfaces of the permeable layer. It is then found that similarity solutions assuming their proper virtual origins can approximately predict the changes in the plume width at the interfaces. The vicinity of the interface is understood as a region of transition between the similarity solutions. The numerical analysis adopts the Beavers-Joseph slip boundary condition with the slip coefficient a estimated as α=1/√(ε), where ε is the prosity of the porous layer. A fairly good agreement between the experimental and numerical results confirms the validity of the slip coefficient a=1/√(ε).

Process of Forming a Turbulent Thermal Plume in a Density-Stratified Turbulent Flow between Parallel Flat Plates.

Processes of forming an organized turbulent thermal plume are investigated in unstable stratified and in combined stably and unstably stratified turbulent flows between parallel flat plates by conditional sampling measurement, in which temperature fluctuation near the wall is used as the detecting signal. The turbulent thermal plume is found to form in an unstable region of both combined and unstable stratified flows. The width of turbulent thermal plume increases downstream in developing unstablystratified turbulent flow. In combined stratified flow, however, development of the thermal plume is suppressed because the stably stratified flow existed over the unstable flow.

A modeling study of the long-range transport of Kosa using particle trajectory methods

The long-range transport of dust in east Asia is analyzed using a conditioned-particle approach. A one-dimensional K–E model with algebraic stress is used to describe the boundary layer dynamics at various locations along the transport path. The turbulent flow fields created using the boundary layer model are used to simulate the transport and dispersion of dust during a spring storm in China. The particle movement studies in the resultant fields indicate that the turbulence is effective in dispersing the plume in the initial few hours after the emissions take place. The boundary layer shear starts dominating the plume dispersion at later times. The calculated plume growth rate is greater while traveling over land surface than when over the sea. This results from the greater strength of the diurnal oscillation and the larger boundary layer shear over the land surface compared to the sea. Calculated plume widths and concentrations are in good agreement with the limited available observations. DOI: 10.1034/j.1600-0889.1993.t01-4-00003.x

Time-Averaged Temperature Characteristics in Transition and Turbulent Regions of the Plume above a Line Heat Source.

The time-averaged temperature characteristics of the turbulent transition and turbulent regime of the thermal plume above a line heat source are elucidated experimentally under the condition without meandering. The time-averaged temperature on the midplane in the turbulent and transitional region can be expressed by (Tcl-T∞)/(Q/λPr)=2.81Gr-x-1/3 and (TCl-T∞)/(Qh(0)/λPr)=2.67×102Gr-x-1/2. When the heat rate is larger than 27.8W/m, the turbulent plume shows self-preservation and the temperature at a given height and halfwidth of the plume can be expressed by Gr-x1/3(T-T∞)/(Q/λPr)=2.81{1 + cos (ξπ/36)}/2 and δ/(λv2Pr/gβQ)1/3=18.0Grx2/15. The plume width increases abruptly in the transitional region.