Frans de Rooij, February 1995
Program in Environmental Fluid Dynamics
Department of Mechanical and Aerospace Engineering
Arizona State University
Tempe AZ 85287
Fluid Dynamics Laboratory
Department of Applied Physics
Eindhoven University of Technology
P.O. Box 513
5600 MB Eindhoven
I am very grateful to my supervisor, professor Gert-Jan van Heijst for giving me the opportunity to go to the United States, a long cherished wish of mine. I am also very thankful that professor H.J.S. Fernando welcomed me in his lab and encouraged my work during our short but always very exciting discussions. I especially thank C.Y. Ching for giving me a hand whenever I encountered a problem and, together with all the people in and around the lab: Eric, Greg, Angelica, Nick, John, Robert, Angela, Theresa, Dave, Dan, Jane, Michael, Ryan, Andrew, Leonard, Sonny, Andy, Mike, and Rob, for the absolutely great time I had in Arizona.
where Vg is the geostrophic wind above the boundary layer, V'max is the fluctuating component of the wind in the direction of V, and V is the mean wind speed at the height 10 m from the ground. The parameter l depends on the background conditions such as surface roughness and surface pressure patterns, and varies over 0.5 < l < 0.9. Because of the high vertical wind shear near the ground, the surface stratification is expected to be unimportant.
The second type of gust producing mechanism is sporadic,
but much more severe and destructive. It is associated with showers and
thunderstorms in which, through a combination of drag due to water loading
and cooling due to evaporation and melting, downdraughts are formed that
transport horizontal momentum to the ground. The lifetime of an individual
downdraught can vary from 2-30 min, and their horizontal scale is 2-5 km.
They may originate at heights from 1500 m to a few kilometers (in severe
convective thunderstorms). When such downdraughts reach the ground, they
impinge on the ground and may spread with very high speeds, typically of
the order of 20 m/s or more. Fujita (1981) called these downdraughts 'microbursts'.
A typical sequence of events occurring during the impingement of a heavy
fluid mass on a ground is represented in figure 1, which is a drawing based
on the present experiments (these events have been reported by previous
workers also - Linden & Simpson 1985). The opposite edges of the front
on either side of the downdraught have velocities in opposite directions
and there is a large differential in velocity across the microburst. In
general, an aircraft entering the leading edge of the microburst front
feels a heavy head wind (which causes the pilot to reduce the engine thrust)
and seconds later it enters strong tailwinds, whence the aircraft loses
its altitude and may plunge on the ground. This horizontal change
of wind velocity is called the "wind shear". Moreover, the appearing vortex
produces large differences in vertical wind velocities close to the leading
edge of the outflow. Several recent aircraft accidents (e.g. flight DL
169 at Dallas-Fort Worth in 1985) have been positively attributed to the
wind shear. In addition, microbursts may cause destruction of buildings;
the violent localized thunderstorm that occurred in Phoenix, Arizona on
September 13th 1994 which destroyed an entire school builing is a good
example for this destructive ability.
Figure 1: Sketch of the early stages of the descent of negatively buoyant fluid from a point source.
When deep convection is present, the maximum gust (microburst) speed is estimated by weather forecasters using the formula (Kershaw & Nakamura 1994)
where it is assumed that the total potential and kinetic energy at the origination height H from the ground is retained in the gust. Here, DQ and qrain are the anomaly of potential temperature and the rain mixing ratio, Q is the average potential temperature of the downdraught and V is the average speed at height H. In most cases V(H) dominates, and hence the weather forecasters use Vmax=Vg in estimating wind shear, assuming that the microbursts originate above the atmospheric boundary layer (e.g. U.K. Meteorological Office). However, it has been found that this latter formula can underpredict the maximum gusts by a factor two.
The aim of the present work is to further study the nature and properties of axi-symmetric buoyant fluid masses descending on solid surfaces. Several such studies have been previously reported, but have been done by releasing buoyant fluid masses on the ground (Linden & Simpson 1985) or by releasing an isolated buoyant blob of heavy fluid (a thermal) from a certain height above the surface (Lundgren et al. 1992). In the present work, we model the downdraughts as a continuous buoyancy source (a plume) located at a certain height above the surface, in view of the facts that the velocity of the descending air might contribute to the strength of the outflow and that the descending first front of the downdraught might reach the ground and start spreading before the buoyancy supply is terminated at the originating point. A point source is used because three-dimensional effects, such as vortex stretching, are supposed to be important. The impingement of these buoyant fluid masses is studied in a still environment as well as in turbulent surroundings.
In the next paragraph the experimental arrangement is described and in §3 the results are presented. A comparison with atmospheric microbursts can be found in §4 and §5 contains the conclusions.
A glass tube with an inside diameter d = 3.97 mm was placed vertically in the middle of the tank at an adjustable height. Via a plastic hose and two control valves this tube was connected to a constant-level tank system containing a saline solution. The fluid level in this system was kept constant at 1.47 m above the bottom plate of the experimental tank.
Figure 2: The experimental apparatus.
Before each experiment the volume flux Q from the glass tube was set with the first control valve and measured by determining the amount of fluid flowing from it during a few minutes. The density of the water in the tank r0 and of the saline fluid r1 were measured with a refractometer. The initial buoyancy or reduced gravity was calculated using g'=g(r1-r0)/r0, the buoyancy flux using B=g'Q and the momentum flux M using M=Q2/¼pd2. The ranges of Q and r1 used were 0.8*10-6 m3/s < Q < 3.5*10-6 m3/s and 1.036*103 kg/m3 < r1 < 1.14*103 kg/m3, which resulted in 3.1*10-7 m4/s3 < B < 3.7*10-6 m4/s3 and 5.2*10-8 < M < 9.9*10-7. After waiting for at least 30 minutes for the motion of water in the experimental tank to die, the orifice of the glass tube was lowered to a level just below the fresh water level. Immediately after that the second control valve was opened and thus the negatively buoyant fluid was released.
To obtain a flow with plume characteristics it is important that the flow is turbulent as well as dominated by buoyancy. For the flow to be turbulent the Reynolds number, defined by Re=rvL/h=4rQ/phd with L the length over which the velocities in the flow change to v, should be sufficiently large. The plume equivalent of the Monin-Obukhov length scale Lm = M3/4B-2/4 is useful to determine whether the buoyancy flux B is dominant over the momentum flux M (Fischer et al. 1979). According to Papanicolau & List (1987) the flow can be regarded as a pure plume if z/Lm>5, with z the vertical distance to the end of the glass tube.
An additional set of experiments was performed in which the momentum flux from the orifice was considerably higher to induce a non-negligible initial momentum. This yielded a turbulent jet rather than a plume. In these experiments the density of the jet r1 was 1.076*103 kg/m3, the volume flux Q » 27*10-6 m3/s and the outflow height was chosen in the range 7.0 cm < H < 16.2 cm.
The impingement was visualized and measured using three different techniques.
During the first series of experiments the flow was visualized by adding fluorescine dye to the saline solution and illuminating a central slice of the experimental tank from below with a 1000 W lamp. The flow in this vertical cross-section was recorded with a videocamera. The speed of the radial spreading as well as the height of the leading vortex were measured directly from the TV-screen. During some runs pictures were taken with a photocamera.
Secondly buoyancy measurements were made using Laser Induced Fluorescence (LIF). About 22*10-6 mass percent fluorescine dye was added to the saline solution and a 200 mW laser with a rotating beamshifter were used to produce a vertical light sheet in the middle of the tank. A filter was mounted on the videocamera to screen out the reflected light and record only the fluorescence. The molecular diffusivity of salt differs from that of fluorescine, but on the time-scale of the experiments only the dilution due to entrainment of ambient fluid is important. Therefore the amount of fluorescent light can be used to obtain the salinity, using fluorescence measurements of solutions with a known concentration as a reference. The videorecords were analysed with the "DigImage" processing software.
Lastly a series of experiments was carried out to obtain an estimate for the velocities within the flow. A few dozen particles were placed in the middle of the experimental tank, right below the glass tube, with the aid of a thin plastic cylinder that was carefully removed after the fluid had calmed down. The diameter of the particles was about 0.8 mm and the density of the water in the experimental tank r0 was matched to their density: 1.042*103 kg/m3. When the particles get entrained their velocity is supposed to be equal to the velocity of the surrounding fluid. The particles in the flow were illuminated through the slit in the bottom with a 400 W lamp and their motion was recorded with a videocamera. The "DigImage" software was used to do the particle tracking on the video records.
To induce the background turbulence during the second set of experiments this plate was roughened by gluing a layer of rocks (average volume 39.7 mm3, smallest dimension between 2.36 and 4.76 mm) to the upper surface. The average density of these stones on the plate was 1 stone per 28 mm2. Using those dimensions the characteristic mesh length m can be estimated to be 4 mm. A series of experiments was performed with the plume impinging on the rough bottom to investigate the influence of the roughness on the flow. Fluorescine dye and laser sheet illumination were used to visualize the flow.
Subsequently the rough plate was connected to a mechanism which oscillated it with a frequency w between 1 and 4 Hz and a stroke S of 2.1 cm. The edges of the plate were wedge-shaped to reduce boundary effects. An Acoustic Doppler Velocimetry (ADV) probe was used to determine the characteristics of the turbulence, such as Root Mean Square (RMS) velocities in three directions. Again using fluorescine dye and laser sheet illumination for visualization the impact of the plume in the presence of background turbulence was studied.
The plume character seems to be established well before the plume reaches the bottom of the tank. Calculation of Lm leads to values between 0.7 cm and 2.8 cm, and since H was varied between 17.7 cm and 28.5 cm the condition z/Lm>5 is indeed satisfied at the bottom for all experiments. The effective plume height H* is defined as the height of the virtual origin of the plume, which is the point were a pure buoyant plume without any initial momentum should originate to yield the same flow in the far field. As can be seen in the photographs in figure 3, H* is a bit smaller than the height of the water level H.
As soon as the plume reaches the bottom the negatively buoyant fluid starts spreading radially along the bottom and a strong leading vortex ring is formed (figure 3a). The initial horizontal speed of the leading front is much higher than the final speed it achieves after travelling a short distance. During the short period of its formation the diameter of the vortex core increases quickly until it reaches a value that remains more or less constant. The vorticity clearly intensifies for a short while due to vortex stretching. The vortex seems to ride on top of a thin layer of the dense fluid, from which it entrains fluid. After this first vortex several smaller vortices appear, mostly not forming a nicely closed ring. Some of them get entrained by the first vortex and merge with it (figure 3c, right-hand side), thus providing the leading vortex the fluid it needs to keep its core diameter constant while increasing its major radius. Eventually the leading vortex seems to lag behind a bit from the current front. The horizontal extent of the vortex core increases and the vortex character becomes less obvious.
In the steady state for continuous flux from the plume
new vortices keep emerging from the impinging plume and spreading radially,
sometimes merging to form larger vortices. Thereby they form a zone of
expanding vortex rings that was also observed by Linden & Simpson (1990).
At the outer end of this zone of rings is the region where the front of
the current behaves like a gravity current. The evolution of the flow is
sketched in figure 1.
Figure 3: Sequential photographs of the impingement of a plume on a flat solid surface.
B=5.93*10-7 m4/s3, H=22.0 cm. The photographs were taken at a 2 s interval.
Figure 4: Radius of simulated microburst versus time after impact.
B=3.06*10-7 m4/s3, H=22.6 cm, H*=20.8 cm.
From the radius-versus-time graphs both the initial and the final velocities were calculated by fitting a straight line to the appropriate data points using a least-squares algorithm.
If it is assumed that the flow evolution is governed by the buoyancy flux B and H*, then it is possible to write the radius of the current R at any time t as
where f1, f2, ... are functions. A plot of R/H* versus t/(H*4/B)1/3 is shown in figure 5a, and an enlargement of the initial radius is given in figure 5b. Apparently, this scaling works well, although figure 6 shows that the data is somewhat scattered. A force fitted spreading law through (0,0) shows that the frontal velocity in the initial phase Vi may be written as
The velocity of the second spreading phase is also amenable to similar scaling, because the input parameters for the problem are B and H*. Figure 7 shows that the final spreading velocity Vf can be written as
Figure 5a: Radius versus time for several microbursts, scaled with parameters B and H*.
Figure 5b: Enlargement of the initial radius versus time for several microbursts, scaled with parameters B and H*.
Figure 6: Initial spreading velocity calculated with a linear least-squares fit versus velocity scale from parameters B and H*.
Figure 7: Final spreading velocity calculated with a linear least-squares fit versus velocity scale from parameters B and H*.
It is worthy to discuss the findings (4) and (5) vis-a-vis to the formula (2). Since V=0 for the present case (2) reduces to
where g' is the initial buoyancy. Plots of Vi and Vf against Ö(2g'H*) are shown in figures 8a,b. Apparently this scaling does not give neat results, probably because g' is not a good scaling parameter for the horizontal outflow due to dilution in the downflow.
Figure 8a: Initial spreading velocity versus maximum velocity estimated using energy conservation.
Figure 8b: Final spreading velocity versus maximum velocity estimated using energy conservation.
To further check the utility of (2), several experiments were done with somewhat larger exit velocities from the nozzle. In these experiments H < Lm so that the buoyant fluid mass emanating from the nozzle now takes the form of a buoyant jet, rather than a plume. According to (2), the velocity of the microburst, after its impingement on the surface, should be
where Up is the exit velocity at the nozzle. The initial and final velocities measured are compared in figure 9a,b. The measured initial propagation velocity of the front of the current is about 15 times smaller than the maximum speed predicted by formula (7).
Figure 9a: Initial spreading velocity scaled with estimated maximum velocity versus nozzle exit velocity factor.
Figure 9b: Final spreading velocity scaled with estimated maximum velocity versus nozzle exit velocity factor.
where t' is the time measured after the impingement. This height was evaluated by direct measurements from the video records of the flow and the results for several runs are shown in figure 10. Note that in each experiment this vortex seems to approach a constant height hf, and in figure 11 we see that hf/H*»(0.20±0.03).
Figure 10: Height of first vortex versus time for several simulated microbursts, scaled with parameters B and H*.
Figure 11: Final height of the first vortex versus the height of the virtual plume origin.
where C1, C2, ... are constants.
To get the buoyancy in the plume as a function of vertical distance from the virtual plume origin z the density along the center line of the plume was averaged over a period of 10 seconds, starting 2 seconds after the impact of the plume. Six experimental runs, with buoyancy fluxes B ranging from 4.8*10-7 to 1.0*10-6 m4/s3 and effective plume heights H* of 0.11 and 0.14 m, were processed in this way. The data of these runs were plotted in a log-log plot according to g'(z)/(B2/3/H*5/3) versus z/H*, in figure 12. The first few centimeters from the orifice the buoyancy is more or less constant. This was also observed by Shabbir & George (1994). At larger distances from the nozzle the buoyancy approximately decreases as z-5/3 down to about 1 or 2 centimeters above the bottom. The constant C1 in equation 9b was evaluated at large z/H* and the value is 21±4. This is quite a bit larger than the value of 9.1 ± 0.5 measured by Rouse (1952) and the value of 14.3 measured by Papanicolaou & List (1988). Maybe the measured effective plume height H* was a bit too high, which could also account for the somewhat steeper slopes in figure 12, or the entrainment of ambient fluid started somewhat later due to the laminar entrance length.
Figure 12: 10 s time averaged buoyancy in the center line of the plume versus distance from virtual plume origin, scaled with parameters B and H*.
In studies of the evolution of the leading vortex, its dilution is of interest to evaluate the entrainment of ambient fluid into the vortex. If one assumes that the height of the vortex h is constant and that only ambient fluid with density r0 gets entrained, applying mass conservation gives
which can be solved to yield
a dimensional analysis shows that the buoyancy in the vortex can be written in the form
The maximum value of the 1s time averaged mean density along a vertical column in a rectangular window containing the leading vortex was measured. The maximum column mean is mostly located at the column that crosses the vortex centre and therefore provides a reasonable estimate of the density in the vortex core. The data of several runs are plotted on a linear plot in figure 13a and on a log-log plot in figure 13b. The amount of scatter in these graphs is large, but the buoyancy in the vortex core clearly decreases as the vortex expands.
Figure 13a: Maximum values of the 1 s time averaged mean density along vertical columns and horizontal lines across the leading edge vortex versus time.
Figure 13b: A log-log plot of the same data scaled with parameters B and H*.
The tracking of on the average 25 particles was performed for the first 20 seconds after the impact of the plume. After these particle paths had been established, the velocities were calculated and extrapolated to a grid across the leading vortex. In figure 14 the results of averaging this velocity grid for one run from 15.5 till 16.5 seconds after the impact are displayed. The maximum absolute value of the velocity Vmax within the leading vortex is plotted versus the time after impact in figure 15. The expected initial increase of Vmax due to vortex intensification can not be seen because just after impact no valid data can be obtained. If we look at the values of Vmax/(B/H*)1/3, which range from 1.8 initially to 0.8 in the final phase of the flow, it is clear that the vorticity in the current can induce velocities near the ground that are much higher than the propagation velocity of the front of the current.
Figure 14: The velocity field in the leading edge vortex 16 s after impact, calculated from particle tracking data. B=7.99*10-7 m4/s3, H=21.2 cm.
Figure 15: The maximum absolute value of the velocity in the leading edge vortex versus time, scaled with parameters B and H*. H* was estimated from H and previous experiments. Both curves are an average of several experiments with the same parameter values.
where the stroke S was equal to 2.1 cm in all experiments. The RMS velocity in vertical direction sz was 20 to 50 percent smaller than sx. Up to 7 cm above the bottom sx is independent of height, whereas sz increases only slightly with height.
The photographs of the flow evolution in figure 16 show that the presence of background turbulence inhibits the formation of a strong leading vortex. Immediately after impact some vorticity seems to roll up into a vortex ring, but the clear vortex intensification observed in the earlier experiments does not occur. The horizontally spreading dense fluid entrains much ambient fluid and the height of the outflow is higher than without background turbulence. Even though there is no clear leading vortex, a bulge at the front of the current can be observed. Billows are formed and shed in upward and backward direction relative to the front. They do not merge to form stronger vorticity elements as in the case without background turbulence.
Figure 16: Sequential photographs of the impingement of a plume on an oscillating rough surface inducing background turbulence. B=2.69*10-6 m4/s3, H=13.2 cm, S=2.1 cm, w=3.11 s-1. The photographs were taken at a 1s interval.
The spreading speed of the front is initially quite high, but it soon slows down to a more or less constant final velocity. A constant initial speed lasting a few seconds and a clear transition to a slower final speed, that were observed in the experiments without the turbulence, are not present. Figure 17a,b shows the scaled initial and final spreading speeds versus the Froude number, in casu sx/(B/H*)1/3. The initial velocity decreases somewhat with increasing Froude number. This effect is a bit stronger for the final velocity. However, the spreading speed of the current is never smaller than the RMS turbulence velocity, which can be seen most clearly in figure 17b.
The experiments with the plume impinging on the non-oscillating rough bottom showed that the roughness itself does not really affect the flow. The corresponding data are shown in figures 17a,b slightly shifted to the right relative to the experiments on the flat bottom, at sx»0.
Figure 17a: Initial spreading velocity scaled with parameters B and H* versus Froude number.
Figure 17b: Final spreading velocity scaled with parameters B and H* versus Froude number.
The so-called DFW downburst that has been studied in extense by Fujita (1986) showed the same characteristics as our impinging plumes. A strong leading-edge vortex, intensified by the stretching of vortex lines due to lateral divergence, was observed. After that first vortex a second and a third vortex were seen spreading outward, just as in the present experiments.
Figure 18: Sketch of the flow of air at a strongly localized thunderstorm.
Figure 19: Doppler radar plots of horizontal radial wind velocities at several altitudes in a thunderstorm above Phoenix, Arizona on August 12, 1994. Beam centerline elevations (ELEV) of 0.5, 2.4, 4.3 and 9.9 degrees correspond to altitudes of approximately 1.5, 4.5, 7 and 15 km. The radar station is located north-east of this storm. Inbound radial velocities [knots] are depicted in shades of blue/green, outbound radial velocities in shades of yellow/red.
Courtesy National Weather Service, Phoenix Office.
After the first vortex new vortices appear. They all propagate radially outward. The initial as well as final spreading velocity of the front of the current scale with (B/H*)1/3. The linearity of the dependence is most clear for the final velocity. The spreading velocities in the presence of background turbulence seem to decrease with increasing Froude number, but more experiments need to be done to establish this dependency.
The agreement with the formula used by weather forecasters is not very good. Measured spreading velocities are at least ten times smaller than the predicted maximum velocities. Particle tracking experiments showed that velocities within the current are almost twice as high as the spreading velocity. Still, the initial buoyancy g' is probably not a good scaling parameter.
Additional measurements on the leading edge vortex showed that its height is approximately constant in the final phase of the flow and that its buoyancy decreases as 1/r or slower. The velocities within this vortex also decrease with increasing r.
Quantitative comparison with atmospheric microbursts is difficult because no data are available on the volume flux of cold air from thunderstorms. Qualitatively the experiments show a good agreement with the available meteorological data.
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