Author:
CAULFIELD C. P.,WOODS ANDREW W.
Abstract
We examine the turbulent gravitational convection which develops
above a point
source of buoyant fluid in a stably stratified environment in which the
buoyancy
frequency varies with height according to
N2=N2s
(z/zs)β. This
generalizes the
classical model of turbulent buoyant plumes rising through uniform and
uniformly
stratified environments originally developed by Morton et al.
(1956). By analogy,
the height of rise of a plume with initial buoyancy flux Fs
has the form Hp=
Apεp−1/2Fs1/4Ns−3/4hp
(λ, β) where εp is the entrainment
constant
for plume motion, Ap
is an O(1) constant, and the non-dimensional plume height,
hp is a function of
&λ=Apεp−1/2Fs1/4Ns−3/4/zs
and β.In the case β>0, the stratification becomes progressively stronger
with height, and
so plumes are always confined within a finite distance above the origin.
Furthermore,
the non-dimensional height of rise h decreases with λ.
In contrast, in the case β<0, the
stratification becomes progressively weaker with height, and so the non-dimensional
plume height increases monotonically with λ. For slowly decaying
stratification,
β>−8/3, the motion is confined within a finite distance
above the source.
However, for each value of β with β<−8/3, there
is a critical value
λc(β) such that for λ<λc
a plume is confined to a region near the source while for
λ[ges ]λc the motion
is unbounded. In the unbounded case, the motion asymptotes to the solution
for
a buoyant plume rising through a uniform environment, with asymptotic buoyancy
flux F∞(λ)<Fs.
We show that in the limiting case λ=λc,
dividing bounded and
unbounded motion, as z→∞ the plume asymptotes to a
new similarity solution
of the second kind which describes the motion of a plume in a non-uniformly
stratified environment. These similarity solutions are unstable in the
sense that small
perturbations to the initial conditions result in very different behaviour
far from the
source.Analogous results for an instantaneous release of buoyant fluid from
a point
source, which forms a thermal, are also presented. The model is applied
to describe
the motion of plumes and thermals in the upper ocean and in naturally ventilated
buildings since in both cases the stratification is typically non-uniform.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
57 articles.
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