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MECHANICAL AND ENVIRONMENTAL ENGINEERING
LABORATORY
PLUME DYNAMICS
MAE126A/171A
Professor Paul Linden
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Figure 1: Plumes rising in the atmosphere.
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DEFINITION OF A TURBULENT PLUME
A turbulent plume is the flow above a sustained source of
buoyancyFor example the flow above a chimney from which hot gases are
being discharged
Gas is released at a volume flow rate Qs at a temperature Ts abovethe temperature Ta of the surrounding air. (The subscript s refers
to the stack or source.)
These gases will rise, partly as a result of the upward momentum
of the flow from the stack, but mainly because the gas is buoyant
compared to the surrounding air.
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Relation between density and temperature of a gas
As the temperature increases gas molecules become more energetic
and their mean-free-paths increase. Consequently, the volume
containing a given number of molecules (equivalently, a given
mass of gas) increases and the density decreases. Hence the density
mass/volume,
decreases.
Perfect gas equation
Many gases, including air are well described by the perfect gas
equation = p/RT,
where p is the pressure, T Kelvin is the temperature and R is the
universal gas constant.
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Entrainment
Entrainment is a process whereby fluid is engulfed into a flow. In aturbulent flow, entrainment occurs by engulfing the surrounding
fluid by the turbulent eddies. This is a random are inherently
unpredictable process.
Entrainment mixes the plume fluid with the surrounding fluid
reducing the concentration of pollutants in the plume.
e.g. flow from a car exhaust otherwise you would breathe CO at
a concentration that would kill you as you walk along a street.
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In a uniform environment,
as a plume rises it entrains cooler air, so that the gas within the
plume gets cooler
the buoyancy force decreases as the plume rises, and the
plume slows down
additional fluid is carried upwards by the plume so the
volume flux Q(z) increases with height
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REDUCED GRAVITY
The buoyancy of the emitted gases is defined in terms of the
reduced gravity g
of the gas which is defined by
g ga s
s,
wheres is the density of the fluid leaving the stack,
a is thedensity of the surrounding air, and g = 9.81 ms2 is the
acceleration of gravity.
The dimensions of reduced gravity
gdim = LT2.
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Using the perfect gas equation
= p/RT,
the reduced gravity g can be determined in terms of temperature.
g = p/RTa p/RTsp/RTs
,
which after rearranging gives
g gTs
TaTa
,
where the subscripts have the same meaning as above.
All temperatures are in Kelvin, so, e.g. the reduced gravity of gases
released at 177C into air at 27C is given by
g = 9.81150
300= 4.9ms2.
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BUOYANCY FLUX
The buoyancy flux B of the plume is defined by
B gQ,
where Q is the volume flow rate (m3
s1
).
Hence, in the above example, if the flow rate Q = 10 m3s1, then
B = 49m4s3.
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In almost all practical cases, B is large enough that the flow is
turbulent. (This is even the case for the plume above a cigarette.)The flow inside the plume is unsteady and has randomness.
Dynamically it contains vorticity, while the surrounding stationary
air is irrotational (and has no vorticity).
As a plume rises it entrains ambient air through a process called
turbulent entrainment. It is not possible to describe or calculate
this entrainment process exactly, and so we model it as discussed
below.
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Figure 3: The buoyancy flux into across the two horizontal planes is thesame otherwise the mass within the volume will increase or decrease.
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THE STRUCTURE OF THE PLUME
For a steady plume one with properties that do not change with
time the buoyancy flux B is constant.
Self similarity
A flow is said to be self similar if the structure of the flow is
geometrically the same at all locations.
e.g. similar triangles are essentially the same shape but at different
scales.
Self similarity is an important concept in physics, as it arises from
situations where systems evolve without remembering their
initial states. For example, if you look at a snowflake and thensmaller and smaller portions of it, the structure is reproduced at
each scale. Often such shapes are fractal, in that their dimension is
non-integer.
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A plume is an example of a self similar flow, since the structure ofthe flow is the same at different distances above the source. The
width b(z) increases with distance z from the source, but when the
properties are rescaled using b as the scale they are the same.
The properties of the plume depend on
the (constant) buoyancy flux B
the distance z above the source
The dimensions of the parameters are
Bdim = L4T3,
zdim = L.
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Self similarity means that the vertical velocity can be written in the
following form
w(r, z) = w0(z)f1(r/b),
where w0 is the vertical velocity on the centerline r = 0 of the
plume, and the function f1(r/b) describes the dependence of the
velocity on the distance r from the center.
To determine the behavior of the centerline velocity, we seek acombination ofB and z that has dimensions of velocity (from the
above decomposition the function f is dimensionless). Obviously,
this leads to
w0 = C1B1/3z1/3,
where C1 is a (dimensionless) constant.
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Similarly the reduced gravity
g(r, z) = g0(z)f2(r/b),
and dimensional analysis gives
g0
= C2B2/3z5/3.
Finally, the width b(z) of the plume is
b = C3z.
Thus the mean shape of the plume is a cone, and the radiusincreases linearly with height.
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From these results we see that the volume flux in the plume
Q(z) = 2
0
w(r, z)rdr,
= C4B1/3z5/3,
and is an increasing function of height.
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Details of the buoyancy and velocity fields
The profiles of the buoyancy and velocity across the plume are
only known from experimental measurements. Measurements of
the time-average of the density and velocity at different distances rfrom the plume centerline at a given vertical distance from the
source are found to be well approximated by the Gaussian curves
g(r, z) = g0
(z)er2/b2 ,
w(r, z) = w0(z)er2/b2 ,
where the subscript zero refers to values measured along the
centerline and b is the plume width.
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THE FILLING BOX
In an enclosed space, such as a room, the plume adds heat
continuously and the air within the space increases continually
provided there are no heat losses. This section investigates how
this increase in temperature occurs.
Figure 4: Schematic of the early stage of a filling box.
temperature
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the initial flow of the air in the plume rises and spreads out as
a hot layer across the ceiling.
with time this warm layer descends as more air is carriedupwards in the plume
at this later stage the plume is entraining air in this layer that is
warmer than that originally in the room.
Consequently, the air arrives at the ceiling hotter than it did at the
start. This process continues so that the plume arriving at the
ceiling is always warmer than at a previous time. Hence, the air
spreading across the ceiling gets hotter with time, and so thetemperature of the warm layer is greatest at the top.
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Figure 5: Later stage of a filling box showing the establishment of a
stable density stratification.
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A fluid which has the warmest, and less dense fluid, at the top is
called stably stratified.
Eventually the initial warm layer descends to the floor, since the
ambient air is continually entrained into the plume by entrainment
and carried to the ceiling. This is called the filling box mechanism.
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In practice it is not necessary to have a completely closed space for
this process to occur. Provided the air is stably stratified, the plume
rise will stop at some height. This is because, although the plumeis initially warmer than the atmosphere, and therefore rises, as a
result of entrainment, the air in the plume gets cooler as it rises.
If the atmosphere is stably stratified, the surrounding air increases
in temperature with height this is called an inversion. So with
height the plume is getting cooler and the air around it is getting
warmer. In most cases, the plume reaches a height where these two
temperatures are equal, and the plume has zero buoyancy. Theplume stops rising and spreads out horizontally.
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If there are many plumes each competing for surrounding air to
entrain the horizontal flow is constrained in the same way as it is
by vertical walls in an enclosed space. Thus in a situation where
there are multiple plumes in an inversion such as occurs, e.g. inLos Angeles the filling box mechanism will recirculate the
pollutants in the plumes and bring them down to ground level.
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On a clear night, the ground cools by long wave radiation which
carries the heat to space. Thus the air near the ground becomescooler than that above, so that it is stably stratified. Since this is
opposite to the usual case found during the day, this situation is
known as an inversion.
This intense radiative cooling is why deserts are cold at night, even
when it is hot during the day, and why in moist atmospheres (such
as coastal regions) there is condensation on cars etc. at night.
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The experiments
The objectives of the experiments are
1. To measure the properties of a turbulent plume
2. To verify self-similarity and determine the profiles of buoyancy
3. To observe the filling box mechanism
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The plume will be set up by a source of salt water in a tank of fresh
water. Since salt water is denser than fresh water the plume isnegatively buoyant, and so the source is placed at the top of the
tank. In this case the flow is simply reversed in direction compared
to a plume rising from a chimney stack.
The fluid equations do not distinguish between gases and liquids
provided they are Newtonian. The only difference is in the values
of the molecular properties the viscosity and the diffusion
coefficient .
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Figure 6: The tank and the peristaltic pump for the salt supply.
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Figure 7: The traverse holding the conductivity probe.
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Figure 8: A close-up of the conductivity probe.
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Figure 9: The probe electronics and the probe-pump.
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1. Week 1 Calibration
Objective
To calibrate the conductivity probe and determine the plumebuoyancy flux
Method
The plume buoyancy flux is given by B = gQ at the source.Calibrate the peristaltic pump to determine the flowrate Q, and
determine g of an initial salt solution.
The conductivity probe is calibrated against g
by placing it in saltsolutions of different strengths.
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Week 2 Plume measurements
Objective
To measure the radial profile of density at 3 different verticallocations beneath the plume source.
To test for self-similarity and determine the unknown constant C2.
MethodUse the traverse to place the probe tip at various locations on a
horizontal cut through the plume. Measure the conductivity at
each location and determine the mean and rms of the signal. Use
the results to fit the theoretical curves to the data, and determine
C2.
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REFERENCES
LINDEN, P.F. 2000 Convection in the environment. Perspectives in
Fluid Mechanics, Eds. Batchelor, G.K, Moffatt, H.K. & Worster, M.G,
Cambridge University Press, pp. 287343.
TURNER, J.S. 1973 Buoyancy effects in fluids. Cambridge University
Press.