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Title:  Investigations On High Rayleigh Number Turbulent Free Convection 
Authors:  Puthenveettil, Baburaj A 
Advisors:  Arakeri, Jaywant 
Submitted Date:  Jun2004 
Publisher:  Indian Institute of Science 
Abstract:  High Rayleigh number(Ra) turbulent free convection has many unresolved
issues related to the phenomenology behind the flux scaling, the
presence of a mean wind and its effects, exponential probability
distribution functions, the Prandtl number dependence and the nature
of near wall structures. Few studies have been conducted in the high
Prandtl number regime and the understanding of near wall coherent
structures is inadequate for $Ra > 10^9$. The present thesis deals
with the results of investigations conducted on high Rayleigh
number turbulent free convection in the high Schmidt number(Sc)
regime, focusing on the role of near wall coherent structures.
We use a new method of driving the convection using concentration
difference of NaCl across a horizontal membrane between two tanks to
achieve high Ra utilising the low molecular diffusivity of NaCl. The
near wall structures are visualised by planar laser induced
fluorescence. Flux is estimated from transient measurement of
concentration in the top tank by a conductivity probe. Experiments
are conducted in tanks of $15\times15\times 23$cm (aspect ratio,AR =
0.65) and $10\times10\times 23$cm (AR = 0.435). Two membranes of
0.45$\mu$ and 35$\mu$ mean pore size were used. For the fine
membrane (and for the coarse membrane at low driving potentials), the
transport across the partition becomes diffusion dominated, while the
transport above and below the partition becomes similar to unsteady
non penetrative turbulent free convection above flat horizontal
surfaces (Figure~\ref{fig:schem}(A)). In this type of convection,
the flux scaled as $q\sim \Delta C_w ^{4/3}$,where $\Delta C_w$ is
the near wall concentration difference, similar to that in Rayleigh 
B\'{e}nard convection . Hence, we are able to study turbulent free
convection over horizontal surfaces in the Rayleigh Number range of
$\sim 10^{10} 10 ^{11}$ at Schmidt number of 602, focusing on the
nature and role of near wall coherent structures. To our knowledge,
this is the first study showing clear images of near wall structures
in high Rayleigh Number  high Schmidt number turbulent free
convection.
We observe a weak flow across the membrane in the case of the coarser
membrane at higher driving potentials (Figure \ref{fig:schem}(B)).
The effect of this through flow on the flux and the near wall
structures is also investigated. In both the types of convection the
near wall structure shows patterns formed by sheet plumes, the common
properties of these patterns are also investigated. The major
outcomes in the above three areas of the thesis can be summarised as
follows
\subsection*{Primary findings}
\label{sec:primaryfindings}
\subsubsection*{High Ra  Sc turbulent free convection }
\label{sec:hrc}
The nondimensional flux was similar to that reported by
Goldstein\cite{gcs} at Sc of 2750. Visualisations show that the near
wall coherent structures are line plumes. Depending on the Rayleigh
number and the Aspect ratio, different types of large scale flow cells
which are driven by plume columns are observed. Multiple large scale
flow cells are observed for AR = 0.65 and a single large scale flow
for AR= 0.435. The large scale flow create a near wall mean shear,
which is seen to vary across the cross section. The orientation of the
large scale flow is seen to change at a time scale much larger than
the time scale of one large scale circulation
The near wall structures show interaction of the large scale flow with
the line plumes. The plumes are initiated as points and then gets
elongated along the mean shear direction in areas of larger mean
shear. In areas of low mean shear, the plumes are initiated as points
but gets elongated in directions decided by the flow induced by the
adjacent plumes. The effect of near wall mean shear is to align the
plumes and reduce their lateral movement and merging. The time scale
for the merger of the near wall line plumes is an order smaller than
the time scale of the one large scale circulation. With increase in
Rayleigh number, plumes become more closely and regularly spaced.
We propose that the near wall boundary layers in high Rayleigh number
turbulent free convection are laminar natural convection boundary
layers. The above proposition is verified by a near wall model,
similar to the one proposed by \cite{tjfm}, based on the similarity
solutions of laminar natural convection boundary layer equations as
Pr$\rightarrow\infty$. The model prediction of the non dimensional
mean plume spacing $Ra_\lambda^{1/3}~=~\lambda /Z_w~=~91.7$  where
$Ra_\lambda$ is the Rayleigh number based on the plume spacing
$\lambda$, and $Z_w$ is a near wall length scale for turbulent free
convection  matches the experimental measurements. Therefore, higher
driving potentials, resulting in higher flux, give rise to lower mean
plume spacing so that $\lambda \Delta C_w^{1/3}$ or $\lambda q^{1/4}$ is
a constant for a given fluid.
We also show that the laminar boundary layer assumption is consistent
with the flux scaling obtained from integral relations. Integral
equations for the Nusselt number(Nu) from the scalar variance
equations for unsteady non penetrative convection are derived.
Estimating the boundary layer dissipation using laminar natural
convection boundary layers and using the mean plume spacing relation,
we obtain $Nu\sim Ra^{1/3}$ when the boundary layer scalar dissipation
is only considered. The contribution of bulk dissipation is found to
be a small perturbation on the dominant 1/3 scaling, the effect of
which is to reduce the effective scaling exponent.
In the appendix to the thesis, continuing the above line of reasoning,
we conduct an exploratory reanalysis (for $Pr\sim 1$) of the Grossman
and Lohse's\cite{gl} scaling theory for turbulent Rayleigh  B\'enard
convection. We replace the Blasius boundary layer assumption of the
theory with a pair of externally forced laminar natural convection
boundary layers per plume. Integral equations of the externally forced
laminar natural convection boundary layer show that the mixed
convection boundary layer thickness is decided by a $5^{th}$ order
algebraic equation, which asymptotes to the laminar natural convection
boundary layer for zero mean wind and to Blasius boundary layer at
large mean winds.
\subsubsection*{Effect of wall normal flow on flux and near wall structures}
\label{sec:effectwallnormal}
For experiments with the coarser($35\mu$) membrane, we observe three
regimes viz. the strong through flow regime
(Figure~\ref{fig:schem}(b)), the diffusion regime (Figure
\ref{fig:schem}(a)), and a transition regime between the above two
regimes that we term as the weak through flow regime.
At higher driving potentials, only half the area above the coarser
membrane is covered by plumes, with the other half having plumes below
the membrane. A wall normal through flow driven by impingement of the
large scale flow is inferred to be the cause of this (Figure
\ref{fig:schem}(b)). In this strong through flow regime, only a single
large scale flow circulation cell oriented along the diagonal or
parallel to the walls is detected. The plume structure is more
dendritic than the no through flow case. The flux scales as $\Delta
C_w^n$, with $7/3\leq n\leq 3$ and is about four times that observed
with the fine membrane. The phenomenology of a flow across the
membrane driven by the impingement of the large scale flow of strength
$W_*$, the Deardorff velocity scale, explains the cubic scaling. We
find the surprising result that the nondimensional flux is smaller
than that in the no through flow case for similar parameters.
The mean plume spacings in the strong through flow regime are larger
and show a different Rayleigh number dependence visavis the no
through flow case. Using integral analysis, an expression for the
boundary layer thickness is derived for high Schmidt number laminar
natural convection boundary layer with a normal velocity at the wall.
(Also, solutions to the integral equations are obtained for the
$Sc\sim 1$ case, which are given as an Appendix.) Assuming the
gravitational stability condition to hold true, we show that the plume
spacing in the high Schmidt number strong through flow regime is
proportional to $\sqrt{Z_w\,Z{_{v_i}}}$, where $Z{_{v_i}}$ is a length
scale from the through flow velocity. This inference is fairly
supported by the plume spacing measurements
At lower driving potentials corresponding to the transition regime,
the whole membrane surface is seen to be covered by plumes and the
flux scaled as $\Delta C_w^{4/3}$.
The nondimensional flux is about the same as in turbulent free
convection over flat surfaces if $\frac{1}{2}\Delta C $ is assumed to
occur on one side of the membrane. This is expected to occur in the
area averaged sense with different parts of the membrane having
predominance of diffusion or through flow dominant transport. At very
low driving potentials corresponding to the diffusion regime, the
diffusion corrected non dimensional flux match the turbulent free
convection values, implying a similar phenomena as in the fine
membrane.
\subsubsection*{Universal probability distribution of near wall structures}
\label{sec:univprobdistr}
We discover that the probability distribution function of the plume
spacings show a standard log normal distribution, invariant of the
presence or the absence of wall normal through flow and at all the
Rayleigh numbers and aspect ratios investigated. These plume
structures showed the same underlying multifractal spectrum of
singularities in all these cases. As the multifractal curve indirectly represents the processes by which
these structures are formed, we conclude that the plume structures are created by a common
generating mechanism involving nucleation at points, growth along
lines and then merging, influenced by the external mean shear.
Inferring from the thermodynamic analogy of multifractal analysis, we
hypothesise that the near wall plume structure in turbulent free
convection might be formed so that the entropy of the structure is
maximised within the given constraints. 
URI:  http://hdl.handle.net/2005/140 
Appears in Collections:  Mechanical Engineering (mecheng)

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